Reserve prices in repeated auctions

Abstract

I consider a model of repeated auctions in which the distribution of bidders’ values is only known to the bidders and the seller attempts to learn this distribution to inform her choice of reserve prices in the future. I find that in any equilibrium bidders will shade their bids to act as if their values are drawn from a lower distribution than they actually are. The bid shading may be so severe that the seller would prefer to simply commit to setting the reserve price that would be optimal if bidders’ values were drawn from the lowest possible distribution to eliminate the incentive for bidders to shade their bids.

Appendix

Lemma 1

For small $\epsilon > 0$ and any r, $\delta$, and $\theta$, $\int _{r}^{r + \epsilon } (v - r) f(v + \delta |\theta )dv = \frac{\epsilon ^2}{2} f(r + \delta | \theta ) + o(\epsilon ^2)$.

Proof

By taking the first-order Taylor series expansion of $f(v|\theta )$, we see that$\int _{r}^{r + \epsilon } (v - r) f(v + \delta |\theta ) \; dv = \int _{r}^{r + \epsilon } (v - r) (f(r + \delta |\theta ) + o(1)) \; dv = \frac{\epsilon ^2}{2} (f(r + \delta | \theta ) + o(1)) = \frac{\epsilon ^2}{2} f(r + \delta | \theta ) + o(\epsilon ^2)$. $\square$

Proof of Theorem 1

Note that if the bidder always bids truthfully in the first period, then the seller will be able to infer the exact value of $\theta$ from the first-period outcomes because there will be a one-to-one mapping between the various possible values of $\theta$ and the fraction of auctions in which the bidder makes a bid greater than or equal to the reserve price. Thus the seller will be able to set a reserve price in the second period that is the optimal reserve price when facing a bidder whose value is drawn from the cumulative distribution function $F(\cdot | \theta )$ for some known value of $\theta$. The optimal such reserve price in the second period, $r_{2}(\theta )$, is an increasing function of $\theta$ satisfying $r_{2}(\theta ) = \beta \theta$ for some $\beta > 0$.

Now suppose that the true value of $\theta$ is close to $\overline{\theta }$. Also suppose that whenever the bidder has a true value v between $r_{1}$ and $r_{1} + \epsilon$ for some small $\epsilon > 0$ that the bidder deviates by bidding less than $r_{1}$ instead of bidding truthfully. Making this change will cost the bidder $\int _{r_{1}}^{r_{1} + \epsilon } (v - r_{1}) f(v|\theta ) \; dv$ in the first period but it will mean that the fraction of auctions in which the bidder makes a bid greater than or equal to the reserve price is only $1 - F(r_{1} + \epsilon | \theta )$ rather than $1 - F(r_{1} | \theta )$. Thus the seller will infer that the true value of $\theta$ is equal to the $\tilde{\theta }$ satisfying $F(r_{1} | \tilde{\theta }) = F(r_{1} + \epsilon | \theta )$.

Now $\int _{r_{1}}^{r_{1} + \epsilon } (v - r_{1}) f(v|\theta ) \; dv = \frac{\epsilon ^2}{2} f(r_{1} | \theta ) + o(\epsilon ^2)$ by Lemma 1, so making this change will cost the bidder $\frac{\epsilon ^2}{2} f(r_{1} | \theta ) + o(\epsilon ^2)$ in the first period. We also know that $F(r_{1} | \tilde{\theta }) = F((\theta /\tilde{\theta }) r_{1} | \theta )$. Thus the value of $\tilde{\theta }$ satisfying $F(r_{1} | \tilde{\theta }) = F(r_{1} + \epsilon | \theta )$ is the value of $\tilde{\theta }$ for which $F((\theta /\tilde{\theta }) r_{1} | \theta ) = F(r_{1} + \epsilon | \theta )$ or the value of $\tilde{\theta }$ for which $(\theta /\tilde{\theta }) r_{1} = r_{1} + \epsilon$, meaning $\tilde{\theta } = \theta \frac{r_{1}}{r_{1} + \epsilon } = \theta \frac{1}{1 + \epsilon /r_{1}} = \theta (1 - \epsilon /r_{1}) + o(\epsilon )$, where the last equality follows from taking a first-order Taylor series expansion of $\frac{1}{1 + \epsilon /r_{1}}$. Since the reserve price in the second period satisfies $r_{2}(\theta ) = \beta \theta$ for some $\beta > 0$, it then follows that if the bidder deviates in the manner given in the previous paragraph, then the reserve price in the second period will be equal to $\beta \tilde{\theta } = \beta \theta (1 - \epsilon /r_{1}) + o(\epsilon )$ rather than $\beta \theta$.

Thus if the bidder deviates in the manner suggested, then the bidder’ssecond-period payoff will be $\int _{\beta \tilde{\theta }}^{\infty } (v - \beta \tilde{\theta }) f(v|\theta ) \; dv$ rather than $\int _{\beta \theta }^{\infty } (v - \beta \theta ) f(v|\theta ) \; dv$. Now $\int _{\beta \tilde{\theta }}^{\infty } (v - \beta \tilde{\theta }) f(v|\theta ) \; dv = \int _{\beta \theta }^{\infty } (v - \beta \tilde{\theta }) f(v|\theta ) \; dv + \int _{\beta \tilde{\theta }}^{\beta \theta } (v - \beta \tilde{\theta }) f(v|\theta ) \; dv$, and the second of these integrals satisfies $\int _{\beta \tilde{\theta }}^{\beta \theta } (v - \beta \tilde{\theta }) f(v|\theta )\; dv = \int _{\beta \tilde{\theta }}^{\beta \tilde{\theta } (1 + \epsilon /r_1)}(v - \beta \tilde{\theta }) f(v|\theta ) \; dv = o(\epsilon )$ by Lemma 1. This in turn implies that$\int _{\beta \tilde{\theta }}^{\infty } (v - \beta \tilde{\theta }) f(v|\theta ) \; dv = \int _{\beta \theta }^{\infty } (v - \beta \tilde{\theta }) f(v|\theta ) \; dv + o(\epsilon )$, so $\int _{\beta \tilde{\theta }}^{\infty } (v - \beta \tilde{\theta }) f(v|\theta ) \; dv$ exceeds $\int _{\beta \theta }^{\infty } (v - \beta \theta ) f(v|\theta ) \; dv$ by $\int _{\beta \theta }^{\infty } \beta (\theta - \tilde{\theta }) f(v|\theta ) \; dv + o(\epsilon ) = \int _{\beta \theta }^{\infty } \beta \theta \epsilon /r_{1} f(v|\theta )dv + o(\epsilon ) = \beta \theta \epsilon /r_{1} (1 - F(\beta \theta | \theta )) + o(\epsilon )$.

From this it follows that the proposed deviation will increase the bidder’s expected payoff in the second period by an amount $\Theta (\epsilon )$ while only decreasing the bidder’s expected payoff in the first period by an amount $\Theta (\epsilon ^2)$. This in turn implies that the proposed deviation is profitable, and there is no equilibrium in which bidders bid truthfully in the first period. $\square$

Proof of Theorem 2

First note that all strategies in which the bidder makes a bid greater than or equal to the reserve price exactly $y(\theta )$ of the time upon learning the value of $\theta$ are observationally equivalent to the seller. Thus the best possible strategy for the bidder amongst the set of strategies in which the bidder makes a bid greater than or equal to the reserve price exactly $y(\theta )$ of the time upon learning the value of $\theta$ is the strategy that results in the largest first-period payoff for the bidder.

In order for the bidder to maximize his first-period payoff subject to the constraint that the bidder makes a bid greater than or equal to the reserve price exactly $y(\theta )$ of the time, the bidder should make a bid greater than or equal to the reserve price in the $y(\theta )$ auctions where the bidder’s value is highest. Thus if $v^{*}(\theta )$ is chosen to satisfy $1 - F(v^{*}(\theta ) | \theta ) = y(\theta )$, the bidder should follow a strategy of bidding more than the reserve if and only if $v > v^{*}(\theta )$. Since this is equivalent to bidding less than the reserve if and only if $v \le v^{*}(\theta )$, in any equilibrium, the bidder must follow a strategy of bidding less than the reserve if and only if $v \le v^{*}(\theta )$ for some $v^{*}(\theta )$ that the bidder chooses upon learning $\theta$. $\square$

The proof of Theorem 3 breaks up into several smaller steps, which I present below as technical lemmas:

Lemma 2

There exists some $\theta _{1} < \overline{\theta }$ such that if bidders follow a strategy of making a bid greater than or equal to the reserve price a fraction $y(\theta ) = 1 - F(r_{1} | \theta _{1})$ of the time whenever $\theta \in [\theta _{1}, \overline{\theta }]$, then a bidder who learns that $\theta = \overline{\theta }$ prefers to follow this strategy and receive a second-period reserve price that equals the reserve price the seller would set if the seller only knew that $\theta \in [\theta _{1}, \overline{\theta }]$ than to bid truthfully and receive a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta = \overline{\theta }$.

Proof

Suppose that $\theta _{1} = \overline{\theta } - \epsilon$ for some small value of $\epsilon > 0$. Note that if the seller only knows that $\theta \in [\theta _{1}, \overline{\theta }]$, then the seller should choose the second-period reserve price $r_{2}$ in such a way to maximize $r_{2} E[1 - F(r_{2} | \theta ) | \theta \in [\theta _{1}, \overline{\theta }]]$, meaning the seller should choose $r_{2}$ to satisfy $E[1 - F(r_{2} | \theta ) | \theta \in [\theta _{1}, \overline{\theta }]] = r_{2} E[f(r_{2} | \theta ) | \theta \in [\theta _{1}, \overline{\theta }]]$, which is equivalent to choosing $r_{2}$ to satisfy $E[1 - F(\frac{\overline{\theta }}{\theta } r_{2} | \overline{\theta }) | \theta \in [\theta _{1}, \overline{\theta }]]= r_{2} E[\frac{\overline{\theta }}{\theta } f(\frac{\overline{\theta }}{\theta } r_{2} | \overline{\theta }) | \theta \in [\theta _{1}, \overline{\theta }]]$.

Now when $\theta _{1} = \overline{\theta } - \epsilon$ for some small value of $\epsilon > 0$, then $E[\frac{\overline{\theta }}{\theta } f(\frac{\overline{\theta }}{\theta } r_{2} | \overline{\theta }) | \theta \in [\theta _{1}, \overline{\theta }]] = E[(1 + \frac{\delta }{\overline{\theta }})(f(r_{2} | \overline{\theta }) + \frac{r_{2}}{\overline{\theta }} \delta f^{\prime }(r_{2} | \overline{\theta })) + o(\delta ) | \delta \in [0, \epsilon ]]$ since when$\theta = \overline{\theta } - \delta$, taking first-order Taylor series implies that $\frac{\overline{\theta }}{\theta } = \frac{1}{1 - \delta /\overline{\theta }} = 1 + \frac{\delta }{\overline{\theta }} + o(\delta )$ and thus $f(\frac{\overline{\theta }}{\theta } r_{2} | \overline{\theta }) = f(r_{2} | \overline{\theta }) + \frac{r_{2}}{\overline{\theta }} \delta f^{\prime }(r_{2} | \overline{\theta }) + o(\delta )$. Thus we also have $E[\frac{\overline{\theta }}{\theta } f(\frac{\overline{\theta }}{\theta } r_{2} | \overline{\theta }) | \theta \in [\theta _{1}, \overline{\theta }]] = E[f(r_{2} | \overline{\theta }) + \frac{\delta }{\overline{\theta }}(f(r_{2} | \overline{\theta }) + r_{2} f^{\prime }(r_{2} | \overline{\theta })) + o(\delta ) | \delta \in [0, \epsilon ]] = f(r_{2} | \overline{\theta })+ \frac{\epsilon }{2 \overline{\theta }}(f(r_{2} | \overline{\theta }) + r_{2} f^{\prime }(r_{2} | \overline{\theta })) + o(\epsilon )$, where the last equality follows from the facts that $E[\delta | \delta \in [0, \epsilon ]] = \frac{\int _{0}^{\epsilon } \delta g(\overline{\theta } - \delta ) \; d \delta }{G(\overline{\theta }) - G(\overline{\theta } - \epsilon )}$, $G(\overline{\theta }) - G(\overline{\theta } - \epsilon ) = \epsilon g(\overline{\theta }) + o(\epsilon )$ from afirst-order Taylor series expansion of $G(\overline{\theta } - \epsilon )$, and $\int _{0}^{\epsilon } \delta g(\overline{\theta } - \delta ) \; d \delta = \int _{0}^{\epsilon } \delta (g(\overline{\theta }) + o(1)) \; d \delta = \frac{\epsilon ^2}{2} (g(\overline{\theta }) + o(1)) = \frac{\epsilon ^2}{2} g(\overline{\theta }) + o(\epsilon ^2)$, so $E[\delta | \delta \in [0, \epsilon ]] = \frac{\int _{0}^{\epsilon } \delta g(\overline{\theta } - \delta ) \; d \delta }{G(\overline{\theta }) - G(\overline{\theta } - \epsilon )} = \frac{\frac{\epsilon ^2}{2} g(\overline{\theta }) + o(\epsilon ^2)}{\epsilon g(\overline{\theta }) + o(\epsilon )} = \frac{\epsilon }{2} + o(\epsilon )$.

Similarly, we have $E[F(\frac{\overline{\theta }}{\theta } r_{2} | \overline{\theta }) | \theta \in [\theta _{0}, \overline{\theta }]] = E[F(r_{2} | \overline{\theta }) + \frac{r_{2}}{\overline{\theta }} \delta f(r_{2} | \overline{\theta }) + o(\delta ) | \delta \in [0, \epsilon ]] = F(r_{2} | \overline{\theta }) + \frac{r_{2}}{2 \overline{\theta }} \epsilon f(r_{2} | \overline{\theta }) + o(\epsilon )$, where I am again taking first-order Taylor series and using the facts that $\frac{\overline{\theta }}{\theta } = 1 + \frac{\delta }{\overline{\theta }} + o(\delta )$ and $E[\delta | \delta \in [0, \epsilon ]] = \frac{\epsilon }{2} + o(\epsilon )$. From this and the result in the previous paragraph, it follows that $r_{2}$ will satisfy $E[1 - F(\frac{\overline{\theta }}{\theta } r_{2} | \overline{\theta }) | \theta \in [\theta _{1}, \overline{\theta }]] = r_{2} E[\frac{\overline{\theta }}{\theta } f(\frac{\overline{\theta }}{\theta } r_{2} | \overline{\theta }) | \theta \in [\theta _{1}, \overline{\theta }]]$ if and only if$1 - F(r_{2} | \overline{\theta }) - \frac{r_{2}}{2 \overline{\theta }} \epsilon f(r_{2} | \overline{\theta }) + o(\epsilon ) = r_{2} f(r_{2} | \overline{\theta }) + \frac{r_{2} \epsilon }{2 \overline{\theta }}(f(r_{2} | \overline{\theta }) + r_{2} f^{\prime }(r_{2} | \overline{\theta })) + o(\epsilon )$, which holds if and only if $1 - F(r_{2} | \overline{\theta }) - r_{2} f(r_{2} | \overline{\theta }) = \frac{r_{2} \epsilon }{2 \overline{\theta }}(2 f(r_{2} | \overline{\theta }) + r_{2} f^{\prime }(r_{2} | \overline{\theta })) + o(\epsilon )$.

Now let $r_{2}(\overline{\theta })$ denote the optimal reserve price for the seller in the second period if the seller knows that $\theta = \overline{\theta }$. We know that $r_{2}(\overline{\theta })$ satisfies $1 - F(r_{2}(\overline{\theta }) | \overline{\theta }) - r_{2}(\overline{\theta }) f(r_{2}(\overline{\theta }) | \overline{\theta }) = 0$. Furthermore, by taking first-order Taylor series approximations, we know that for values of $r_{2}$ satisfying $r_{2} = r_{2}(\overline{\theta }) - \delta$ for some small $\delta > 0$ that $1 - F(r_{2} | \overline{\theta }) - r_{2} f(r_{2} | \overline{\theta }) = 1 - F(r_{2}(\overline{\theta }) | \overline{\theta }) - r_{2}(\overline{\theta }) f(r_{2}(\overline{\theta }) | \overline{\theta })+ \delta [2 f(r_{2}(\overline{\theta }) | \overline{\theta }) + r_{2}(\overline{\theta }) f^{\prime }(r_{2}(\overline{\theta }) | \overline{\theta })] + o(\delta ) = \delta [2 f(r_{2}(\overline{\theta }) | \overline{\theta }) + r_{2}(\overline{\theta }) f^{\prime }(r_{2}(\overline{\theta }) | \overline{\theta })]+ o(\delta )$. Thus $r_{2} = r_{2}(\overline{\theta }) - \delta$ will satisfy $1 - F(r_{2} | \overline{\theta }) - r_{2} f(r_{2} | \overline{\theta }) = \frac{r_{2} \epsilon }{2 \overline{\theta }}(2 f(r_{2} | \overline{\theta }) + r_{2} f^{\prime }(r_{2} | \overline{\theta })) + o(\epsilon )$ if and only if $\delta [2 f(r_{2}(\overline{\theta }) | \overline{\theta }) + r_{2}(\overline{\theta }) f^{\prime }(r_{2}(\overline{\theta }) | \overline{\theta })] + o(\delta )= \frac{r_{2} \epsilon }{2 \overline{\theta }}(2 f(r_{2} | \overline{\theta }) + r_{2} f^{\prime }(r_{2} | \overline{\theta })) + o(\epsilon )$. And since taking first-order Taylor series guarantees that $2 f(r_{2} | \overline{\theta }) + r_{2} f^{\prime }(r_{2} | \overline{\theta }) = 2 f(r_{2}(\overline{\theta }) | \overline{\theta }) + r_{2}(\overline{\theta }) f^{\prime }(r_{2}(\overline{\theta }) | \overline{\theta }) + o(1)$ when $r_{2} = r_{2}(\overline{\theta }) - \delta$, this previous expression holds if and only if $\delta [2 f(r_{2}(\overline{\theta }) | \overline{\theta }) + r_{2}(\overline{\theta }) f^{\prime }(r_{2}(\overline{\theta }) | \overline{\theta })] + o(\delta ) = \frac{r_{2} \epsilon }{2 \overline{\theta }}(2 f(r_{2}(\overline{\theta }) | \overline{\theta }) + r_{2}(\overline{\theta }) f^{\prime }(r_{2}(\overline{\theta }) | \overline{\theta })) + o(\epsilon )$, which in turn holds if and only if $\delta = \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )$.

This in turn implies that the value of $r_{2} = r_{2}(\overline{\theta }) - \delta$ satisfying $1 - F(r_{2} | \overline{\theta })- r_{2} f(r_{2} | \overline{\theta }) = \frac{r_{2} \epsilon }{2 \overline{\theta }}(2 f(r_{2} | \overline{\theta }) + r_{2} f^{\prime }(r_{2} | \overline{\theta })) + o(\epsilon )$ is $r_{2} = r_{2}(\overline{\theta }) - \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )$. From this it follows that if the seller only knows that $\theta \in [\theta _{1}, \overline{\theta }]$ in the second period when $\theta _{1} = \overline{\theta } - \epsilon$ for some small value of $\epsilon > 0$, then the seller will choose a reserve price $r_{2}$ in the second period given by $r_{2} = r_{2}(\overline{\theta }) - \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )$, where $r_{2}(\overline{\theta })$ is the optimal second-period reserve price if the seller knows that $\theta = \overline{\theta }$.

Now if a bidder who learns that $\theta = \overline{\theta }$ follows a strategy of making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{1})$ of the time, then this bidder will make a bid greater than or equal to the reserve if and only if $v > v^{*}(\overline{\theta })$, where $v^{*}(\overline{\theta })$ is chosen to satisfy $1 - F(r_{1} | \theta _{1}) = 1 - F(v^{*}(\overline{\theta }) | \overline{\theta })$ or $F(r_{1} | \overline{\theta } - \epsilon ) = F(v^{*}(\overline{\theta }) | \overline{\theta })$ when $\theta _{1} = \overline{\theta } - \epsilon$. But $F(r_{1} | \overline{\theta } - \epsilon ) = F(\frac{\overline{\theta }}{\overline{\theta } - \epsilon } r_{1} | \overline{\theta })$, so this $v^{*}(\overline{\theta })$ must satisfy$v^{*}(\overline{\theta }) = \frac{\overline{\theta }}{\overline{\theta } - \epsilon } r_{1} = \frac{r_{1}}{1 - \epsilon /\overline{\theta }} = r_{1}(1 + \frac{\epsilon }{\overline{\theta }}) + o(\epsilon )$, where the last equality follows from taking a first-order Taylor series expansion of $\frac{r_{1}}{1 - \epsilon /\overline{\theta }}$.

Thus by making a bid greater than or equal to the reserve if and only if$v > v^{*}(\overline{\theta }) \equiv r_{1}(1 + \frac{\epsilon }{\overline{\theta }}) + o(\epsilon )$ the bidder will lose $\int _{r_{1}}^{r_{1}(1 + \epsilon /\overline{\theta }) + o(\epsilon )} (v - r_{1}) f(v | \overline{\theta }) \; dv= \frac{r_1^2 \epsilon ^2}{2\overline{\theta }^2} f(r_{1} | \overline{\theta }) + o(\epsilon ^2)$ in first-period utility relative to bidding truthfully, where the equality follows from Lemma 1. And the bidder will then face a second-period reserve price of $r_{2} = r_{2}(\overline{\theta }) - \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )$ rather than $r_{2}(\overline{\theta })$, so the bidder’s second-periodpayoff will be $\int _{r_{2}(\overline{\theta }) - \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )}^{\infty } (v - r_{2}(\overline{\theta }) + \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )) f(v | \overline{\theta }) \; dv$ rather than $\int _{r_{2}(\overline{\theta })}^{\infty } (v - r_{2}(\overline{\theta })) f(v | \overline{\theta }) \; dv$. But $\int _{r_{2}(\overline{\theta }) - \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )}^{\infty } (v - r_{2}(\overline{\theta }) + \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )) f(v | \overline{\theta }) \; dv = \int _{r_{2}(\overline{\theta })}^{\infty } (v - r_{2}(\overline{\theta }) + \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )) f(v | \overline{\theta }) \; dv + \int _{r_{2}(\overline{\theta }) - \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )}^{r_{2}(\overline{\theta })} (v - r_{2}(\overline{\theta }) + \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )) f(v | \overline{\theta }) \; dv$. And the second of these integrals is $o(\epsilon )$ by Lemma 1. Thus $\int _{r_{2}(\overline{\theta }) - \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )}^{\infty } (v - r_{2}(\overline{\theta }) + \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )) f(v | \overline{\theta }) \; dv = \int _{r_{2}(\overline{\theta })}^{\infty } (v - r_{2}(\overline{\theta }) + \frac{r_{2} \epsilon }{2 \overline{\theta }} + o(\epsilon )) f(v | \overline{\theta }) \; dv + o(\epsilon )$, which exceeds $\int _{r_{2}(\overline{\theta })}^{\infty } (v - r_{2}(\overline{\theta })) f(v | \overline{\theta }) \; dv$ by $\int _{r_{2}(\overline{\theta })}^{\infty } \frac{r_{2} \epsilon }{2 \overline{\theta }} f(v | \overline{\theta }) \; dv + o(\epsilon ) = \frac{r_{2} \epsilon }{2 \overline{\theta }}(1 - F(r_{2}(\overline{\theta }) | \overline{\theta })) + o(\epsilon )$.

But this means that if $\theta _{1} = \overline{\theta } - \epsilon$ for some small value of $\epsilon > 0$, then a bidder who learns that $\theta = \overline{\theta }$ prefers to meet the reserve price a fraction $1 - F(r_{1} | \theta _{1})$ of the time and receive a second-period reserve price that equals the reserve the seller would set if the seller only knew that $\theta \in [\theta _{1}, \overline{\theta }]$ than to bid truthfully and receive asecond-period reserve price that equals the reserve the seller would set if the seller knew that $\theta = \overline{\theta }$. Bidding truthfully would cost the bidder $\Theta (\epsilon )$ in second-period utility while only bringing $\Theta (\epsilon ^2)$ in additional first-period utility. Thus for sufficiently small $\epsilon > 0$, setting $\theta _{1} = \overline{\theta } - \epsilon$ will indeed meet the desired properties set forth in the statement of this lemma. $\square$

For my next result, I inductively define $\theta _{k}$ as follows. I let $\theta _{0} \equiv \overline{\theta }$ and for all $k > 0$, I then set $\theta _{k}$ to be the largest value of $\theta \in [\underline{\theta }, \theta _{k-1})$ such that a bidder who learns that $\theta = \theta _{k-1}$ is indifferent between bidding truthfully in the first period and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k-1}, \theta _{k-2})$ and making a bid greater than or equal to thefirst-period reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time and receiving asecond-period reserve price that equals the reserve price that the seller would set if the seller knew that $\theta \in [\theta _{k}, \theta _{k-1})$ if such a $\theta$ exists and set $\theta _{k} = \underline{\theta }$ otherwise. I first note the following result:

Lemma 3

There exists some finite m for which $\theta _{m} = \underline{\theta }$.

Proof

Note that there exists some $\epsilon > 0$ such that $|\theta _{k-1} - \theta _{k}| > \epsilon$ for all $k > 0$ such that $\theta _{k} > \underline{\theta }$. The same reasoning used to prove Lemma 2 shows that if $\theta _{k}$ is arbitrarily close to $\theta _{k-1}$, then a bidder who learns that $\theta = \theta _{k-1}$ will strictly prefer making a bid greater than or equal to the first-period reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k}, \theta _{k-1})$ to bidding truthfully in the first period and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k-1}, \theta _{k-2})$. Thus there must be some small $\epsilon > 0$ such that $|\theta _{k-1} - \theta _{k}| > \epsilon$ for all $k > 0$ such that $\theta _{k} > \underline{\theta }$. But this in turn means that it must be some finite m for which $\theta _{m} = \underline{\theta }$. $\square$

Throughout the remainder of this appendix, I let m denote the smallest positive integer for which $\theta _{m} = \underline{\theta }$. Given this definition, I next obtain the following result.

Lemma 4

If $\theta = \theta _{m-1}$, there exists some fraction $y(\theta )$ such that a bidder is indifferent between making a bid greater than or equal to the first-period reserve price a fraction $y(\theta )$ of the time and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m}, \theta _{m-1})$ and bidding truthfully in the first period and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m-1}, \theta _{m-2})$.

Proof

We know that if $y(\theta ) = 1 - F(r_{1} | \theta _{m})$, then the bidder must prefer making a bid greater than or equal to the reserve price a fraction $y(\theta )$ of the time and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m}, \theta _{m-1})$ to bidding truthfully in the first period and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m-1}, \theta _{m-2})$. The definition of $\theta _{m}$ implies there is no $\theta _{m} \in [\underline{\theta }, \theta _{m-1})$ for which the bidder prefers bidding truthfully in the first period and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m-1}, \theta _{m-2})$ to making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{m})$ of the time and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m}, \theta _{m-1})$.

We also know that if $y(\theta ) = 0$, then the bidder must prefer bidding truthfully in the first period and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m-1}, \theta _{m-2})$ to making a bid greater than or equal to the reserve price a fraction $y(\theta )$ of the time and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m}, \theta _{m-1})$. By bidding truthfully in the first period rather than never making a bid greater than or equal to the reserve price, the bidder increases his first-period payoff by an amount equal to the amount by which his first-period payoff would increase if he always bid truthfully, but the reserve price were reduced from $\infty$ to $r_1$. But if the seller knew that $\theta \in [\theta _{m-1}, \theta _{m-2})$, then the seller would set a second-period reserve price no greater than $r(\theta _{m-2})$, the optimal reserve price if the seller knew that $\theta = \theta _{m-2}$, and if the seller knew that $\theta \in [\theta _{m}, \theta _{m-1})$, then the seller would set a second-period reserve price no less than $r(\theta _{m})$, the optimal reserve price if the seller knew that $\theta = \theta _{m}$.

Thus if $y(\theta ) = 0$, then by bidding truthfully in the first period rather than making a bid greater than or equal to the reserve price a fraction $y(\theta )$ of the time, the bidder increases his first-period payoff by the amount by which his payoff would be increased if he bid truthfully and the reserve price were reduced from $\infty$ to $r_1$ but he only decreases his second-period payoff by no more than the amount by which his payoff would be decreased if the reserve price increased from $r(\theta _{m})$ to $r(\theta _{m-2})$. From this it follows that if $y(\theta ) = 0$, then the bidder prefers bidding truthfully in the first period and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m-1}, \theta _{m-2})$ to making a bid greater than or equal to the reserve price a fraction $y(\theta )$ of the time and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m}, \theta _{m-1})$.

By combining this result with the result in the first paragraph of this proof, it follows from the intermediate value theorem that there is some $y(\theta ) \in [0, 1 - F(r_{1} | \theta _{m})]$ such that a bidder who learns that $\theta = \theta _{m-1}$ is indifferent between making a bid greater than or equal to the first-period reserve price a fraction $y(\theta )$ of the time and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m}, \theta _{m-1})$ and bidding truthfully in the first period and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m-1}, \theta _{m-2})$. $\square$

Proof of Theorem 3

Now suppose that the players make use of the following beliefs and strategies: If $\theta \in [\theta _{k}, \theta _{k-1})$ for some $\theta _{k} > \underline{\theta }$, then the bidder makes a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time. And if $\theta \in [\theta _{m}, \theta _{m-1})$, where $\theta _{m} \equiv \underline{\theta }$, then the bidder makes a bid greater than or equal to the reserve price a fraction $y(\theta )$ of the time, where $y(\theta )$ is chosen in such a way that a bidder who learns that $\theta = \theta _{m-1}$ is indifferent between making a bid greater than or equal to the first-period reserve price a fraction $y(\theta )$ of the time and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m}, \theta _{m-1})$ and bidding truthfully in the first period and receiving a second-period reserve price that equals the reserve price the seller would set if the seller knew that $\theta \in [\theta _{m-1}, \theta _{m-2})$. We know that this construction of the bidder’s strategy is feasible by Lemma 4.

For the seller I assume that if the bidder makes a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time for some $k < m$, then the seller believes that $\theta$ is a random draw from the distribution $G(\cdot | \theta \in [\theta _{k}, \theta _{k-1}))$. If the bidder makes a bid greater than or equal to the reserve price a fraction $y(\theta )$ of the time, where $y(\theta )$ is the fraction of the time that bidders would make a bid greater than or equal to the reserve price if $\theta \in [\theta _{m}, \theta _{m-1})$, then the seller believes that $\theta$ is a random draw from the distribution $G(\cdot | \theta \in [\theta _{m}, \theta _{m-1}))$. Otherwise the seller believes that $\theta = \overline{\theta }$. The seller then sets a second-period reserve price that equals the optimal second-period reserve price given the seller’s beliefs. I now seek to show that these beliefs and strategies are a perfect Bayesian equilibrium:

First note that the seller’s beliefs and strategies are consistent with a perfect Bayesian equilibrium. If the bidder makes a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time for some $k < m$, then the seller knows that $\theta \in [\theta _{k}, \theta _{k-1})$, and the seller’s beliefs are correct. If the bidder makes a bid greater than or equal to the reserve price a fraction $y(\theta )$ of the time, where $y(\theta )$ is the fraction of the time that bidders would make a bid greater than or equal to the reserve price if $\theta \in [\theta _{m}, \theta _{m-1})$, then the seller knows that $\theta \in [\theta _{m}, \theta _{m-1})$, and the seller’s beliefs are again correct. Any other possibility occurs with probability zero, so any seller beliefs are consistent with a perfect Bayesian equilibrium in this case. And the seller’s choice of strategy is clearly consistent with a perfect Bayesian equilibrium since it is always optimal to set a second-period reserve price that equals the optimal second-period reserve price given the seller’s beliefs.

Next I show that the bidder’s strategy is a best response to the seller’s strategy. To do this, I first show that the derivative of a bidder’s utility difference between making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time and obtaining a second-period reserve price of $r_{l}$ and making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{j})$ of the time and obtaining a second-period reserve price of $r_{h}$, where $\theta _{k} < \theta _{j}$ and $r_{h} > r_{l}$, is strictly increasing in $\theta$.

To see this, first note that if the bidder makes a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time, then the bidder follows a strategy of making a bid greater than or equal to the reserve price if and only if $v > v^{*}_{k}(\theta )$, where $v^{*}_{k}(\theta )$ is chosen to satisfy $F(v^{*}_{k}(\theta ) | \theta ) = F(r_{1} | \theta _{k})$. Since $F(v^{*}_{k}(\theta ) | \theta ) = F(\frac{\theta _{k}}{\theta } v^{*}_{k}(\theta ) | \theta _{k})$, it then follows that this $v^{*}_{k}(\theta )$ will also satisfy $\frac{\theta _{k}}{\theta } v^{*}_{k}(\theta ) = r_{1}$ or $v^{*}_{k}(\theta ) = \frac{\theta }{\theta _{k}} r_{1}$. Similarly, if the bidder makes a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{j})$ of the time, then the bidder follows a strategy of making a bid greater than or equal to the reserve price if and only if $v > v^{*}_{j}(\theta )$, where $v^{*}_{j}(\theta ) = \frac{\theta }{\theta _{j}} r_{1}$.

Thus the difference between a bidder’s first-period utility from making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{j})$ of the time and making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time is then $\int _{v^{*}_{j}(\theta )}^{v^{*}_{k}(\theta )} (v - r_{1}) f(v | \theta ) \; dv = \int _{\frac{\theta }{\theta _{j}} r_{1}}^{\frac{\theta }{\theta _{k}} r_{1}} (v - r_{1}) \frac{\theta _{j}}{\theta } f(\frac{\theta _{j}}{\theta } v | \theta _{j}) \; dv = \int _{r_{1}}^{\frac{\theta _{j}}{\theta _{k}}r_{1}} (\frac{\theta u}{\theta _{j}}- r_{1}) f(u | \theta _{j}) \; du$. The derivative of this first-period utility difference with respect to $\theta$ is then strictly positive and independent of $\theta$.

Similarly the difference between a bidder’s second-period utility from making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time and making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{j})$ of the time is then $\int _{r_{l}}^{\infty } (v - r_{l}) f(v | \theta ) \; dv - \int _{r_{h}}^{\infty } (v - r_{h}) f(v | \theta ) \; dv = \int _{r_{l}}^{r_{h}} (v- r_{l}) f(v | \theta ) \; dv + (r_{h} - r_{l})(1 - F(r_{h} | \theta )) = \int _{r_{l}}^{r_{h}} (v - r_{l}) \frac{\theta _{k}}{\theta } f(\frac{\theta _{k}}{\theta } v | \theta _{k}) \; dv+ (r_{h} - r_{l})(1 - F(\frac{\theta _{k}}{\theta } r_{h} | \theta _{k})) = \int _{\frac{\theta _{k} r_{l}}{\theta }}^{\frac{\theta _{k} r_{h}}{\theta }} (\frac{\theta u}{\theta _{k}} - r_{l}) f(u | \theta _{k}) \; du + (r_{h} - r_{l})(1 - F(\frac{\theta _{k}}{\theta } r_{h} | \theta _{k}))$. The derivative of this second-period utility difference with respect to $\theta$ is then $\int _{\frac{\theta _{k} r_{l}}{\theta }}^{\frac{\theta _{k} r_{h}}{\theta }} \frac{u}{\theta _{k}} f(u | \theta _{k}) \; du - \frac{\theta _{k} r_{h}}{\theta ^2} (r_{h} - r_{l}) f(\frac{\theta _{k} r_{h}}{\theta } | \theta _{k}) + \frac{\theta _{k} r_{h}}{\theta ^2} (r_{h} - r_{l}) f(\frac{\theta _{k} r_{h}}{\theta } | \theta _{k})= \int _{\frac{\theta _{k} r_{l}}{\theta }}^{\frac{\theta _{k} r_{h}}{\theta }} \frac{u}{\theta _{k}} f(u | \theta _{k}) \; du$. This derivative is thus strictly positive and decreasing in $\theta$ because $\frac{d}{d\theta } \int _{\frac{\theta _{k} r_{l}}{\theta }}^{\frac{\theta _{k} r_{h}}{\theta }} \frac{u}{\theta _{k}} f(u | \theta _{k}) du = -\frac{\theta _{k} r_{H}^2}{\theta ^3} f(\frac{\theta _{k} r_{H}}{\theta } | \theta _{k}) + \frac{\theta _{k} r_{L}^2}{\theta ^3} f(\frac{\theta _{k} r_{L}}{\theta } | \theta _{k})= \frac{\theta _{k}}{\theta ^3}(r_{L}^2 f(\frac{\theta _{k} r_{L}}{\theta } | \theta _{k}) - r_{H}^2 f(\frac{\theta _{k} r_{H}}{\theta } | \theta _{k})) = \frac{1}{\theta ^2}(r_{L}^2 f(r_{L} | \theta ) - r_{H}^2 f(r_{H} | \theta )) < 0$.

By combining the results in the previous two paragraphs, it follows that thederivative of a bidder’s utility difference between making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{j})$ of the time and obtaining a second-period reserve price of $r_{h}$ and making a bid greater than or equal to the reserve price afraction $1 - F(r_{1} | \theta _{k})$ of the time and obtaining a second-period reserve price of $r_{l}$, where $\theta _{k} < \theta _{j}$ and $r_{h} > r_{l}$ is strictly increasing in $\theta$. Now a bidder with $\theta = \theta _{k-1}$ will be indifferent between making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time and receiving a second-period reserve price equal to the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k}, \theta _{k-1})$ and making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k-1})$ of the time and receiving a second-period reserve price equal to the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k-1}, \theta _{k-2})$. And a bidder with $\theta = \theta _{k}$ will strictly prefer making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time and receiving a second-period reserve price equal to the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k}, \theta _{k-1})$ to making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k-1})$ of the time and receiving a second-period reserve price equal to the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k-1}, \theta _{k-2})$.

By combining the three facts in the previous paragraph it follows that a bidder with $\theta < \theta _{k-1}$ must prefer making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time and receiving a second-period reserve price equal to the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k}, \theta _{k-1})$ to making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k-1})$ of the time and receiving a second-period reserve price equal to the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k-1}, \theta _{k-2})$. Similarly, if $\theta > \theta _{k-1}$, then the bidder prefers making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k-1})$ of the time and receiving a second-period reserve price equal to the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k-1}, \theta _{k-2})$ to making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time and receiving a second-period reserve price equal to the reserve price the seller would set if the seller knew that $\theta \in [\theta _{k}, \theta _{k-1})$.

From this it follows that if a bidder has a type $\theta \in [\theta _{k}, \theta _{k-1})$, then the bidder prefers making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time rather than following any other strategy of making a bid greater than or equal to the reserve price a different fraction of the time $y(\theta )$ that is followed by some other bidder type. Furthermore, a bidder with type $\theta \in (\theta _{k}, \theta _{k-1})$ for some positive integer $k \le m$ has no incentive to deviate by bidding truthfully in the first period because such a deviation would result in a smaller increase in the bidder’s first-period utility than the first-period utility increase a bidder with $\theta = \theta _{k-1}$ would obtain by bidding truthfully and would result in a larger decrease in the bidder’s second-period utility than the second-period utility loss a bidder with $\theta = \theta _{k-1}$ would obtain by bidding truthfully since the bidder with type $\theta \in (\theta _{k}, \theta _{k-1})$ would face a higher reserve price by bidding truthfully than the bidder with $\theta = \theta _{k-1}$. Since the bidder with $\theta = \theta _{k-1}$ is indifferent between bidding truthfully and making a bid greater than or equal to the reserve price a fraction $1 - F(r_{1} | \theta _{k})$ of the time, a bidder with type $\theta \in (\theta _{k}, \theta _{k-1})$ prefers following the strategy I have specified to deviating by bidding truthfully.

Finally, no bidder can profitably deviate to some other (non-truthful) strategy that involves making a bid greater than or equal to the reserve price a different fraction of the time than any bidder would on the equilibrium path. Such a deviation would result in a smaller first-period payoff increase than bidding truthfully, but result in the same second-period payoff loss as truthful bidding because the bidder would face the same second-period reserve price. Since truthful bidding was not a profitable deviation, deviating to some such (non-truthful) strategy also cannot be a profitable deviation.

By combining the results in the previous paragraphs, we see that neither the bidder nor the seller can profitably deviate from the strategies described. Thus there exists an m and a sequence of cutoffs $\theta _{0}, \ldots , \theta _{m}$ satisfying $\underline{\theta } = \theta _{m}< \theta _{m-1}< \cdots< \theta _{1} < \theta _{0} = \overline{\theta }$ such that there is an equilibrium in which the fraction of auctions in which the bidder makes a bid greater than or equal to the reserve price, $y(\theta )$, is independent of $\theta$ for all $\theta \in [\theta _{k}, \theta _{k-1})$. $\square$

Proof of Theorem 4

Suppose each advertiser is following the monotonic biddingstrategy $b(v, \theta )$. Then we know that $y(\theta | b)$ and $R(\theta | b)$ are both non-decreasingfunctions of $\theta$. From this it in turn follows that $y(\theta | b)$ and $R(\theta | b)$ are both continuous in $\theta$ almost everywhere. Thus there is also a positive measure of values of $\theta ^*$ at which $y(\theta | b)$ and $R(\theta | b)$ are both continuous in $\theta$ when $\theta = \theta ^*$ such that either (a) $y(\theta ^* | b) < 1$ or (b) there is a positive measure of values of $\hat{\theta } < \theta ^*$ at which $y(\hat{\theta } | b) = 1$.

At such a value of $\theta ^*$, in the limit as $\theta ^{\prime }$ approaches $\theta ^*$ from below, $y(\theta ^{\prime } | b)$ and $R(\theta ^{\prime } | b)$ become arbitrarily close to $y(\theta ^* | b)$ and $R(\theta ^* | b)$ respectively. Thus for some $\theta ^{\prime }$ sufficiently close to $\theta ^*$, it will be possible for a bidder i to deviate to some other bidding strategy $b_{i}(v, \theta )$ in the first period such that, if $\theta = \theta ^*$ and all other bidders follow the bidding strategy $b(v, \theta )$, then the fraction of the time that some bidder will meet the reserve price is $y(\theta ^{\prime } | b)$ and expected revenue is $R(\theta ^{\prime } | b)$.

Now if following the monotonic bidding strategy $b(v, \theta )$ in the first period would fully reveal the value of $\theta$, then the seller will use a reserve price $r(\theta )$, equal to the optimal reserve when bidders’ values are drawn from the distribution $F(v|\theta )$, in all future periods. Thus from the results in the previous paragraphs, we know there is a positive measure of values of $\theta ^*$ for which if bidder i deviates to some alternative bidding strategy $b_{i}(v, \theta )$, then the reserve price in future periods will be lowered to some $r(\theta ^{\prime }) < r(\theta ^*)$.

This implies that if a bidder deviates to the bidding strategy $b_{i}(v, \theta )$ in the first period, then the bidder will change the reserve prices that bidders face in all future periods in such a way that bidders increase their expected payoffs from each future period by some fixed amount. If $\delta$ is sufficiently close to 1, this will be a profitable deviation, which in turn implies that there is no equilibrium in which the bidders follow monotonic first-period bidding strategies that would fully reveal the value of $\theta$. $\square$

Proof of Theorem 5

Let $r(\theta ^*)$ denote the optimal reserve price that the seller would set in a one-shot auction if the seller knew that $\theta = \theta ^*$ and all bidders bid truthfully. Also suppose that the various players make use of the following beliefs and strategies.

If there is some positive integer k for which $\theta \in (\theta _k, \theta _{k-1}]$, then each bidder follows a strategy of bidding $b(v|\theta ) = \frac{\theta _k}{\theta } v$ as long as in each previous period t, the bidder has received the outcome he would receive if all other bidders were following the same strategy when the reserve price is $r_{t}$. And if in some previous period t a bidder receives a different outcome than the outcome the bidder would receive if all other bidders were bidding $b(v|\theta ) = \frac{\theta _k}{\theta } v$ when the reserve price is $r_{t}$, then each bidder follows a strategy of bidding truthfully in all future periods.

Before the bidders have bid in the first period, the seller believes that $\theta$ is a random draw from the cumulative distribution function $G(\cdot )$. After the bidders have bid in at least one period, as long as the distribution of prices in each previous period was the same as it would have been if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ for some positive integer $k \le m$, then the seller believes that $\theta$ is a random draw from the distribution $G(\cdot | \theta \in (\theta _k, \theta _{k-1}])$. If in the first period the distribution of prices differs from what would result if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ for each positive integer $k \le m$, then the seller believes that $\theta = \overline{\theta }$ with probability 1. And if in the first t periods the distribution of prices was the same as it would have been if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ for some positive integer $k \le m$ and in period $t+1$ the distribution of prices differs from this, then the seller believes that $\theta = \theta _{k-1}$ with probability 1 after period $t+1$.¹⁴

The seller follows a strategy of setting a first-period reserve price equal to the optimal reserve price if for all positive integers $k \le m$, there was a probability $G(\theta _{k-1}) - G(\theta _k)$ that each bidder’s bid was a random draw from the distribution $F(\cdot | \theta _k)$. In periods $t \ge 2$, the seller follows a strategy of setting a reserve price equal to $r(\theta _k)$ as long as the distribution of prices in each previous period was the same as it would have been if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$. If in the first period the distribution of prices differs from what would result if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ for each positive integer $k \le m$, then the seller sets a reserve price equal to $r(\overline{\theta })$ in periods $t \ge 2$. And if in the first t periods the distribution of prices was the same as it would have been if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ for some positive integer $k \le m$ and in period $t+1$ the distribution of prices differed from this, then the seller sets a reserve price of $r(\theta _{k-1})$ after period $t+1$. I show that these beliefs and strategies are a perfect Bayesian equilibrium:

First note that the seller’s beliefs are consistent with a perfect Bayesian equilibrium. Before the bidders have bid in the first period, the seller believes that $\theta$ is a random draw from the cumulative distribution function $G(\cdot )$. Along the equilibrium path, the bidders always follow strategies such that the distribution of prices in each period is the same as it would be if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ if and only if $\theta \in (\theta _k, \theta _{k-1}]$. Thus if the distribution of prices in each previous period was the same as it would have been if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$, then the seller believes that $\theta$ is a random draw from the distribution $G(\cdot | \theta \in (\theta _k, \theta _{k-1}])$. All other circumstances can only arise off the equilibrium path, so any beliefs that do not contradict the seller’s previous beliefs are consistent with a perfect Bayesian equilibrium. In particular, it is consistent for a seller to believe that $\theta = \overline{\theta }$ if there is a deviation in the first period and that $\theta = \theta _{k-1}$ if the distribution of prices in the first t periods was the same as it would have been if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ for some positive integer $k \le m$ and the distribution of prices differed from this in period $t+1$.

Next note that, given these beliefs, a seller cannot profitably deviate from theproposed strategy. It is a best response for the seller to set a first-period reserve price equal to the optimal reserve price if for all positive integers $k \le m$, there was aprobability $G(\theta _{k-1}) - G(\theta _k)$ that each bidder’s bid was a random draw from thedistribution $F(\cdot | \theta _k)$ because this is the distribution from which bids will be made. In periods $t \ge 2$, it is also a best response for the seller to set a reserve price equal to $r(\overline{\theta })$ if the distribution of prices in the first period differed from what would arise if the bidders all made bids drawn from the distribution $F(\cdot | \theta _k)$ for each $k \le m$, because if this happens, then the seller believes that $\theta = \overline{\theta }$ and all bidders will bid truthfully in the future, so it is optimal to set the reserve price $r(\overline{\theta })$ in all future periods.

It is also a best response for the seller to follow a strategy of setting a reserve price equal to $r(\theta _k)$ in any period $t \ge 2$ if the distribution of prices in each previous period was the same as it would have been if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ for the following reason: Under such a history, the seller believes that $\theta \in (\theta _k, \theta _{k-1}]$ with certainty and the bidders will follow a strategy of bidding $b(v|\theta ) = \frac{\theta _k v}{\theta }$ in both this period t and all future periods regardless of what reserve price the seller sets in period t. Thus the distribution of bids that the seller faces in period t and all future periods will be the same distribution the seller would face if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$. Thus the seller’s best response in this case is to set a reserve price of $r(\theta _k)$ as this will optimize the seller’s expected payoff in the current period t and have no effect on the seller’s payoff in future periods.

Finally if in the first t periods the distribution of prices was the same as it would have been if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ for some positive integer $k \le m$, and in period $t+1$ the distribution of prices differed from this, then it is a best response for the seller to set a reserve of $r(\theta _{k-1})$ in all future periods, because if this happens, then the seller believes that $\theta = \theta _{k-1}$ and all bidders will bid truthfully in the future, so it is optimal to set the reserve price $r(\theta _{k-1})$ in all future periods.

To see that the bidders cannot profitably deviate from the proposed strategy, note that if $\theta \in (\theta _k, \theta _{k-1}]$ for some positive integer k and in some previous period t a bidder has received a different outcome than the outcome the bidder would receive in equilibrium, then all other bidders will bid truthfully going forward and the seller will always set the same reserve price going forward, so a bidder’s best response will be to bid truthfully. And if this has not happened in any previous period, and a bidder deviates from the proposed strategy, then by the assumption on the values of $\theta _0, \ldots , \theta _m$ in the theorem, the resulting distribution of prices in that period will differ from what would arise if each bidder’s bid were a random draw from the distribution $F(\cdot | \theta _k)$ for all k, the seller will set a strictly higher reserve price in all future periods than she would have had the bidder not deviated, and all other bidders will bid truthfully in all future periods rather than bidding a fraction of their true value. Thus the bidder will obtain a lower payoff in all future periods by deviating, and if $\delta$ is sufficiently close to 1, this is not a profitable deviation either.

From this it follows that if $\delta$ is sufficiently close to 1, then neither the bidders nor the seller is able to profitably deviate from the proposed strategies. Thus if $\delta$ is sufficiently close to 1, then there is an equilibrium in which if $\theta \in (\theta _k, \theta _{k-1}]$, then each bidder follows a strategy of bidding $b(v|\theta ) = \frac{\theta _k}{\theta } v$ along the equilibrium path. $\square$

Proof of Corollary 2

In the special case where $m = 1$, the condition on the values of $\theta _0, \ldots , \theta _m$ in Theorem 5 is trivially satisfied, so we know from Theorem 5 that there is an equilibrium in which if $\theta \in (\theta _1, \theta _0]$, then each bidder follows a strategy of bidding $b(v|\theta ) = \frac{\theta _1}{\theta } v$, which is equivalent to bidding $b(v|\theta ) = \frac{\underline{\theta } v}{\theta }$. In this case, the seller’s best response is to always set a reserve price equal to $r(\underline{\theta })$, the optimal reserve price that the seller would set in a one-shot auction if the seller knew that $\theta = \underline{\theta }$ and all bidders were bidding truthfully, so such strategies are an equilibrium. $\square$

Proof of Theorem 6

If the bidders follow the equilibrium strategies in Theorem 5 governed by the cutoffs $C = \{\theta _0, \ldots , \theta _m\}$, then for a given value of $\theta$, each bidder makes a bid of $b(v|\theta ) = \frac{\theta _k}{\theta } v$, where $\theta _k$ denotes the largest value in C that is less than $\theta$, and the seller sets a reserve price equal to $r(\theta _k)$ in all periods after the first period. Thus when the bidders follow such equilibrium strategies, the seller’s expected revenue for a given realization of $\theta$ is equal to some function $R(\theta _k(\theta | C))$, where $R(\cdot )$ is a strictly increasing function, and $\theta _k(\theta | C)$ is a function that gives the largest value in C that is less than $\theta$.

Now for any $\theta \in [\underline{\theta }, \overline{\theta }]$, the value of $\theta _k(\theta | C)$ is at least as large as the value of $\theta _k(\theta | C^{\prime })$ when $C^{\prime }$ is a subset of C. Furthermore, since $C^{\prime }$ is a strict subset of C, there is a positive measure of values of $\theta \in [\underline{\theta }, \overline{\theta }]$ for which $\theta _k(\theta | C)$ is strictly greater than $\theta _k(\theta | C^{\prime })$. Thus if bidders follow the equilibrium strategies in Theorem 5 governed by the cutoffs in C, then the seller’s expected revenue for any given value of $\theta$, $R(\theta _k(\theta | C))$, is at least as large as the seller’s expected revenue if bidders follow the equilibrium strategies in Theorem 5 governed by the cutoffs in $C^{\prime }$, $R(\theta _k(\theta | C^{\prime }))$. Furthermore, there is a positive measure of values of $\theta \in [\underline{\theta }, \overline{\theta }]$ for which $R(\theta _k(\theta | C))$ is strictly greater than $R(\theta _k(\theta | C^{\prime }))$. From this it follows that the seller’s revenue will be higher if the bidders follow the equilibrium strategies in Theorem 5 governed by the cutoffs in C than if the bidders follow the corresponding equilibrium strategies governed by the cutoffs in $C^{\prime }$. $\square$

Proof of Theorem 7

If the seller could commit to always setting a reserve price equal to $r(\underline{\theta })$, then the bidders would have a dominant strategy of bidding truthfully in all periods. Thus each bidder would bid more in each period than the bidder does in the equilibrium in Theorem 5 while the seller is setting the same reserve price as in the equilibrium in Theorem 5. From this it follows that the seller’s revenue would be higher if the seller could commit to always setting a reserve price equal to $r(\underline{\theta })$ than it is in the equilibrium in Theorem 5. $\square$