Revenue Equivalence

For a number of differing auction settings, we consider the sale of a single item amongst fixed number of auction participants. It is interesting that under a certain game-theoretic construction all these auctions can be seen to be equivalent.

The auction model which we describe below is called the symmetric independent private values model (SIPV).

An auctioneer has a single, indivisible item for sale. $N$ bidders wish to bid for the item.
Each bidder $i=1,...,N$ believes the items value is $v_i$ euros, where $v_i>0$ .
We assume each value $v_i$ is random variable which is independent of all other values, has cumulative distribution $F(\cdot)$ and has a continuous probability density function $f(\cdot)$ , i.e. values are symmetric and independent.
We assume each bidder $i$ knows his value $v_i$ and the density function $f(\cdot)$ . We assume the bidder does not explicitly know the other bidders values $v_j$ for $j\neq i$ , i.e. values are private.
Each bidder $i$ chooses a bid $b_i=b_i(v_i)$ as a function $v_i$ and $f(\cdot)$ .
From the vector bidders bids $b=(b_i: i=1,...,N)$ , the auctioneer chooses the outcome of the auction: who wins the item and how much each bidder pays.
Given bids $b$ , the function $x=x(b)$ gives the decision of the auctioneer. Here $x(b)\in \{ e_i:i=1,...,N\}$ where $e_i$ , the $i$ th unit vector in ${\mathbb R}_+^N$ , signifies that bidder $i$ has won the item.
Given bids $b$ , the vector $p=p(b)=(p_i(b):i=1,...,N)$ gives the payment given by each bidder.
We call a vector of decisions and payments, $(x,p)$ , an outcome. The function $(x(\cdot),p(\cdot))$ , which map bids to outcomes, is the auction mechanism.
We assume the mechanism auction is symmetric, namely, permuting the entries of the bid vector $b$ will correspondingly permute of the entries of $x(b)$ and $p(b)$ . So, there is not inherent bias towards any bidder.
Given the vectors of valuations $v$ and bids $b$ , the payoff to bidder $i$ is

$u^i(b,v_i)=\mathbf{v}^i\cdot x(b) - p_i(b),$

Here $\mathbf{v}^i\in{\mathbb R}_+^N$ , take value $v_i$ in its $i$ th component and zero in all other components.

Two popular forms of auctions are the English auction, where bidders sequential increase the current highest bid until one bidder remains, and the Dutch auction, where the price of the item is lowered incrementally until one bidder agrees to pay the current price. We will not focus on the dynamics of such auctions. Instead, we simply consider sealed bid auctions. Here bidders to write down their bid and then pass this on to the auctioneer without any communication with other bidders. The sealed bid analogue of the English auction and Dutch auction are the second-price sealed bid auction¹, where highest bid wins the item and pays the price of the second highest bid, and the first-price sealed bid auction, where the highest bid wins the item and pays a price equal to that winning bid. Because the biggest value is, of course, bigger than the second biggest, it is tempting to assert that a first-price auction would generate more revenue that a second price auction. However, this does not account for how bidders will alter their bids to gain a good payoff, and as we see shortly this intuition can be seen to be incorrect. Once again, the notion of equilibrium for our SIPV auction.

A Bayes-Nash Equilibrium is a set of bidding strategies $b(\cdot)=(b_i(\cdot): i=1,...,N)$ such that, given their value, each bidder maximizes their expected payoff

$b_i(v)\in \argmax_{\tilde{b}_i\geq 0} {\mathbb E}\left[ u^i(\tilde{b}_i,b_{-i}(v_{-i}),v) \Big| v_i=v \right]$

$= \argmax_{\tilde{b}_i\geq 0} \int_0^\infty \!\!\!\!\!\hdots \int_0^\infty u^i(\tilde{b}_i,b_{-i}(v_{-i}),v) \times \prod_{j:j\neq i} f(v_j)\times dv_1...dv_{i-1}dv_{i+1}...dv_{N}.$

Here $b_{-i}(v_{-i})=(b_1(v_1),...,b_{i-1}(v_{i-1}),b_{i+1}(v_{i+1}),...,b_N(v_N))$ is the vector with the $i$ th component removed.
The above equilibrium assumes that bidders are risk neutral, that is they are concerned only with maximizing their expected payoff and are not worried with the random variation in that payoff.
For a Nash equilibrium each player was explicitly aware of who she was playing, their strategies and the games payoffs. In this auction setting, each player is not aware of the type of their opponents, in particular, the amount each opponent values the item is unknown. We view this unknown parameter as a random variable (i.e. Bayesian) and then players look for a strategy/bid with the best expected reward. This is a Bayes-Nash Equilibrium.

The following next will aid our analysis of a Bayes-Nash Equilibrium.

Theorem [The Envelope Theorem] For a function $U:{\mathbb R}_+\times {\mathbb R}_+\rightarrow {\mathbb R}_+$ , $(b,v)\mapsto U(b,v)$ , we define

$W(v)=\max_{\tilde{b}} U(\tilde{b},v)\quad\text{and}\quad b(v)\in \argmax_{\tilde{b}} U(\tilde{b},v).$

We suppose that

$\text{1. }\frac{\partial U}{\partial v}(b,v)\text{ exists for all }b.$

$\text{2. } \frac{\partial U}{\partial v}(b(v),v)\text{ is continuous in }v.$ Then

$W(v)=W(0) + \int_0^v \frac{\partial U}{\partial v}(b(y),y)dy.$

Proof: Because, for any $v$ and $v'$ , $W(v) \geq U(b(v'),v)$ and $W(v') \geq U(b(v),v')$ , we have

$\frac{U(b(v),v)-U(b(v),v')}{|v-v'|}$

$\geq \frac{W(v)-W(v')}{|v-v'|}$

$\geq \frac{U(b(v'),v)-U(b(v'),v')}{|v-v'|}.$

Letting $v'\searrow v$ the first inequality and letting $v\searrow v'$ in the second inequality respectively gives²

$\frac{d W}{d v}(v)\geq \frac{\partial U}{\partial v}(b(v),v)\quad\text{and}\quad\frac{d W}{d v}(v')\leq \frac{\partial U}{\partial v}(b(v'),v')$ $\quad\implies\quad \frac{d W}{d v}(v)= \frac{\partial U}{\partial v}(b(v),v).$

Integrating this continuous function gives $W(v)=W(0) + \int_0^v \frac{\partial U}{\partial v}(b(y),y)dy \square$ .

We consider the expected payoff of bidder $i$ , with fixed value $\tilde{v}\in{\mathbb R}_+$ and bid $\tilde{b}\in{\mathbb R}_+$ , and then observe the best expected payoff:

$U^i(\tilde{b},\tilde{v}) := {\mathbb E} \left[ u^i\big(\tilde{b},b_{-i}(v_{-i}),\tilde{v}\big)\right]$

$W^i(\tilde{v}):=\max_{\tilde{b}\geq0} U^i(\tilde{b},\tilde{v}).$

From now on, we assume that the conditions of the Envelope Theorem are satisfied by $U^i(\tilde{b},\tilde{v})$ .

Thrm [Myerson’s Payoff Equivalence Theorem] For a symmetric independent private values auction at Bayes-Nash equilibrium $b(\cdot)$ , expected payoffs satisfy,

$W^i(\tilde{v})-W^i(0)=\int_0^{\tilde{v}} w^i(y) dy$

where $w^i(v)={\mathbb P}(\; i \text{ wins the auction }| v_i=v )$ and, consequently, the expected payments are given by

$\label{RE:payoff} {\mathbb E}[p_i(b(v)) | v_i=\tilde{v}]= -W^i(0) + \tilde{v} w^i(\tilde{v}) - \int_0^{\tilde{v}} w^i(y)dy.$

The crucial point here is that the Bayes-Nash expected payoff $W^i(v)$ is a function of $w^i(\cdot)$ players $i$ ’s chance of winning given his valuation. The function $w^i(\cdot)$ need not depend on the payment structure of the auction. So for all auctions for which $w^i(\cdot)$ is the same, the payoffs for $i$ remain the same.

Proof: By assumption the Envelope Theorem applies to $W^i(v)$ and $U^i(t,v)$ . Thus

$\label{RE:Apply Env} W^i(v)-W^i(0)= \int_0^v \frac{\partial U^i}{\partial v}(b_i(y),y) dy.$

Now, recalling $u^i(b,v)=\mathbf{v}^i\cdot x(b) - p_i(b)$ , for $\tilde{v}$ fixed and for independent random variables $v_{-i}$ , we observe

$\frac{\partial U^i}{\partial v}(b_i,v)={\mathbb E}\left[ \frac{\partial u^i}{\partial v}\big((b_i,b_{-i}(v_{-i})), v\big) \right]$

$= {\mathbb E} \left[ x^i(b_i,b_{-i}(v_{-i}))\right].$

Thus at Bayes-Nash equilibrium, if bidder $i$ has value $v$ then she will bid $b_i(v)$ . So

$\label{RE:diff U} \frac{\partial U^i}{\partial v}\big(b_i(v),v\big)={\mathbb E}\left[ x^i(b(v)) \big| v_i=v \right]=w^i(v).$

In the second inequality above, we note $x^i$ equals $1$ if $i$ wins the auction and equals zero otherwise. Thus, the expected value of $x^i$ is $w^i(y)$ , the probability that bidder $i$ wins the item. Substituting, into gives

$W^i(v)-W^i(0) = \int_0^v w^i(y) dy.$

Observing that $W^i(\tilde{v})= \tilde{v}w^i(\tilde{v})- {\mathbb E}[p_i(b(v)) | v_i=\tilde{v}]$ , from the above equality the equation immediately follows. $\square$

Now we can prove one of the most famous results in auction theory: the Revenue Equivalence Theorem. In very loose terms, the Revenues Equivalence Theorem states that the expected revenue to the auctioneer – that is the expected payments collected from all bidders – is the same for many SIPV auctions. More precisely:

Thrm [Revenue Equivalence Theorem] Consider a symmetric independent private values auction with $N$ bidders where the conditions of the Envelope Theorem are satisfied. The expected revenue of the auctioneer is equal for all auctions where a bidder with value $0$ makes a zero payment and where, at a Bayes-Nash Equilibrium, the item is always awarded to the bidder with the highest valuation.

Numerous assumptions are made in the statement of this result: SIPV, risk-neutral bids, constant number of bidders. Even so, it is striking that a large array of auctions can be seen to be equivalent.

Proof If a bidder with value $0$ makes no payment then $W^i(0)=0$ . If the bidder with the highest value wins then

$w^i(v)= {\mathbb P}( v \geq v_j,\text{ for }j\neq i) = F(v)^{N-1}$

Applying this to equation , we see that $R$ , the expected revenue of the auction, is

$R={\mathbb E}\left[\sum_{i=1}^N p_i(b(v))\right]= N{\mathbb E}\left[ p_i(b(v))\right]$ $=N\int_0^\infty {\mathbb E}\left[ p_i(b(v)) \Big| v_i=\tilde{v}\right] f(\tilde{v}) d\tilde{v}$ $=N\int_0^\infty \left( \tilde{v}F(\tilde{v})^{N-1} - \int_0^{\tilde{v}} F(y)^{N-1} dy \right) f(\tilde{v})dv.$

$\square$

This auction is also sometimes called the Vickery auction.↩
Here we assume the limit $\frac{d W}{d v}$ exists. For more rigor, to show $\frac{d W}{d v}$ exists, you can apply $\liminf$ and $\limsup$ and see that these are equal.↩