Each bidder has a randomly determined private value for one unit of a good. Each submits a bid without knowing the bids of others. The auction ends when all players have submitted bids. The one or more units go to the highest bidders, who pay either their own bids or the highest non-winning bid.
Demonstrates the workings of a private-value sealed-bid auction.
The equilibrium bidding strategy depends on the pricing rule. When winners pay the highest non-winning bid, the weakly dominant strategy is for each player to bid her value. When winners pay their bid, a bidder has an incentive to shade her bid.
The second price private-value sealed-bid auction is expected to generate the same seller revenue as a first price private-value sealed-bid auction.
By default, there is one item on auction each round (Units=1). Each bidder values one unit, with her private value drawn from uniform distribution with the default endpoints Lowest Bidder Value=$10 and Highest Bidder Value=$100. Each bidder draws a new value each period if Periods > 1.
Equilibrium bidding strategy depends on the pricing rule. With Price Rule=First Price, the winning bidder pays her bid. Assuming bidders are risk-neutral (and the uniform distribution is common knowledge), the Bayesian Nash Equilibrium Bid Function is [(N-1)/N]*Buyer Value, where N=Group Size. With default parameters, the Bayesian Nash Equilibrium Bid Function is 5/6*Buyer Value.
With Price Rule=Second Price, the winning bidder pays the second highest bid. Since each player’s bid only affects whether she wins or loses (but not how much she pays if she wins), each player has a weakly dominant strategy to bid her value.
For a multi-unit auction, set Units > 1 (making sure that Units < Group Size). Your options for Price Rule will change. When set to Uniform, each winning bidder pays the highest non-winning bid. For example, if Units=2, then the two winners each pay the third highest bid. When set to Discriminatory, each winning bidder pays her bid.
Setting a Reserve Price sets the minimum price the seller must receive in order to part with an item. For the second-price auction (or uniform price, if multi-unit), if the highest non-winning bid is less than the reserve price, then those bidding at or above the reserve price pay the reserve price.
Describe the private value auction setting as a game with the set of bidders \(N = \{1,2,\cdots,n\}\). For each bidder \(i\), he/she obtains value \(v_i\), which is randomly drawn from a uniformly distributed interval \(v_i \in [\underline{v}, \bar v]\) according to the cumulative distribution function \(F_i(.)\), where \(\underline v\) is the lowest bidder value and \(\bar v\) is the highest bidder value from the configuration setting. We assume that the draw of \(v_i\) is independent across \(i\).
We assume that the distribution function is common knowledge and the true value of \(v_i\) is only known to bidder \(i\). Also, we use the \(n-1\) cumulative distribution functions \(F_j(.)\), \(j\neq i\), to form beliefs about the type \(v_{-i}\) of the other bidders. The payoff of a player who does not get the good is normalized to zero, and the payoff of a bidder who gets the good and pays a price \(p \geq 0\) is \(u_i= v_i-p\).
To complete the derivation of equilibrium bidding strategy of a sealed-bid first-price auction, we shall assume that bidder \(i\)'s equilibrium bid is increasing in his/her own valuation. That is, if \(v_i' > v_i''\), then \(s_i(v_i') > s_i(v_i'')\). It ensures that each bidder's bid function is invertible, which means that bidder \(i\)'s type is given by the inverse function of the bidder's strategy, \(s_i^{-1}(b)\). The probability of \(i\)'s bid is higher than \(j\)'s bid is thus
\(\Pr(b_i > s_i(v_i)) = \Pr(s_j^{-1}(b_i)>v_j)=F_j(s_j^{-1}(b_i)).\)
Based on independent types distribution, the expect payoff of bidder \(i\) in a first-price sealed-bid auction is
\(E_{v_{-i}}[u_i(b_i,s_{-i}(v_{-i});v_i|v_i)] = \Pi_{j\neq i}[F_j(s_j^{-1}(b_i))]\times(v_i-b_i)\).
Under the game setting, bidder \(i\)'s beliefs on others' types is the same, \(\forall i\). The maximization problem is
\(\max_{b\geq 0} E_{v_{-i}}[u_i(b_i,s_{-i}(v_{-i});v_i|v_i)] = [F(s^{*-1}(b))]^{n-1}\times(v_i-b_i)\).
Assuming an interior solution we can write down the first-order condition as follows
\(-[F(s^{*-1}(b))]^{n-1}+(n-1)[F(s^{*-1}(b))]^{n-2}f(s^{*-1}(b))\frac{ds^{*-1}(b)}{db}(v_i-b)=0\).
In order that the profile of bidding strategies \((s_1(.),\cdots,s_n(.)) = (s^*(.),\cdots,s^*(.))\) be a Bayesian Nash Equilibrium, \(b = s^*(v)\) has to be a solution to the above first order condition. Moreover, with \(\frac{ds^{*-1}(b)}{db} = \frac{1}{s^{*'}(s^{-1}(b))}\), the first order condition can be rewritten as
\(-[F(v)]^{n-1}+\frac{(n-1)[F(v)]^{n-2}f(v)(v-b)}{s^{*'}(v)}=0\).
Rewriting the above equation into a differential equation, the symmetric Bayesian Nash Equilibrium Bid Function is
\(s^*(v) = v - \frac{\int_{\underline v}^v[F(x)]^{n-1}dx}{[F(v)]^{n-1}}\).
In our game setting, bidder \(i\) is randomly drawn for a uniform distribution. The equilibrium bidding strategy is therefore
\(S^*(v) = v - \frac{\int^v_0 x^{n-1}dx}{v^{n-1}} = (\frac{n-1}{n})v\).
The payoff function of each bidder \(i\) is
\(u_i(b_i,b_{-i};v_i) = \begin{cases} v_i - b_i^*, &\text{ if } b_i>b_j, \forall j \neq i, \text{ and } b_i \equiv \max_{j\neq i }\{b_j\}; \\0, &\text{ if } b_i \leq b_j, \text{for some } j\neq i. \end{cases}\)
Given that a bidder's action is his/her bid, a bidder's strategy assigns a bid to each of the bidder's types. The strategy for a player \(i\) is a function \(s_i:[\underline v,\bar v]\rightarrow \mathbb{R}_+\) that assigns a nonnegative bid to each of his/her possible valuations. The expected payoff of bidder \(i\) with private valuation \(v_i\), given strategies used by the other bidders \(s_j(.),\ j\neq i\), is
\(E_{v_{-i}}[u_i(b_i,s_{-i}(v_{-i});v_i|v_i)] = \Pr(i \text{ wins and pays } b_i )\times(v_i-b_i)+\Pr(i \text{ loses })\times 0\).
Consider the cases of a bidder who is bidding below his valuation, \(b_i < v_i\). There are three possible cases with respect to the other \(n-1\) bids:
Bidder \(i\) is the highest bidder, in which case \(i\) wins and pays a price \(p < b_i\). Instead of bidding \(b_i\), player \(i\) would have bid \(v_i\), then he/she would still win and pay the same price, so in this case bidding his/her valuation is indifferent to bidding \(b_i\).
The highest bidder \(j\) bids \(b_j^{*}>v_i\), in which case \(i\) loses. Instead of bidding \(b_i\) bidder \(i\) would have bid \(v_i\) then he/she would still lose to \(b^*_j\), so in this case bidding his/her valuation is indifferent to bidding \(b_i\)
The highest bidder \(j\) bids \(b_j^*\) with \(b_i < b_j^*< v_i\), so that \(i\) does not win. If instead bidder \(i\) would have bid \(v_i\), he/she would have won the auction and received a payoff of \(u_i=v_i-b_j^*>0\), making this a profitable deviation, so in this case bidding his/her valuation is strictly better than bidding \(b_i\).
We can also show that bidding bidder \(i\)'s own valuation is also weakly dominant bidding \(b_i >v_i\). The fact that every player has a weakly dominant strategy, \(s_i(v_i) = v_i\), implies that each player bidding his valuation is a Bayesian Nash Equilibrium in weakly dominant strategies.1
The results highlight the Nash Equilibrium and show how close your students were to equilibrium play. Use the Go To: menu to switch between periods.
For each round, there is a table summarizing each group’s performance (Figure 1). With Units = X, the first columns display the sum of the X-highest item values (Optimal Surplus) and the sum of the winners' item values (Surplus).2 The final column shows seller Revenue (highest non-winning price × X). Check the radio button to reveal a figure summarizing a group’s bidding.
By unchecking the first two check-boxes for the Nash Equilibrium Bid Function and Best Fit Line, only markers that indicate the sealed bid and bidder value for each participant remain (Figure 2).
To compare participant choices with the Nash equilibrium, use the Nash Equilibrium Bid Function line by clicking the box adjacent to its legend entry. With the Best Fit Line box checked, a linear trend line is fitted to participant bids and values (Figure 3). This line will coincide with the Bid Function if all students played equilibrium strategies.
Our robot (i.e., an automated player) strategy is the following:
The robot will submit a bid equal to their value multiplied by \(\frac{N-1}{N}\) where \(N\) is the number of bidders in the auction.
Robot submits a bid equal to their value.
1. Tadelis, S. (2013). Game Theory: An Introduction. Princeton University Press.↩
2. Surplus calculations assume the items have no value to the seller. ↩