Heterogeneous bids in auctions with rational and boundedly rational bidders: theory and experiment

Abstract

We present results from a series of experiments that allow us to measure overbidding and, in particular, underbidding in first-price auctions. We investigate the extent to which the amount of underbidding depends on the seemingly innocuous parameters of the experimental setup. To structure our data, we present and test a theory that introduces constant markdown bidders into a population of fully rational bidders. While a fraction of bidders in the experiment can be described by Bayesian Nash equilibrium bids, a larger fraction seems either to use constant markdown bids or to rationally optimise against a population with fully rational and boundedly rational markdown bidders.

Appendices

A. Derivation of the best-response function $\alpha _i^{*}(\alpha _j)$ with markdown bids as given by (4)

With the uniform distribution $f(x_i)=f(x_j)=1$ for $x_i,x_j\in [0,1]$, and CRRA utility, the expected utility function simplifies to

$$\begin{aligned} \text {EU}_{i}(\alpha _{i})=\left\{ \begin{array}{lll} [ 1-\frac{1}{2}\left( \alpha _i-\alpha _j+1\right) ^2]\,\alpha ^r_i &{} &{} \text {if }\max \{0,\alpha _j-1\}\le \alpha _{i}\le \alpha _{j}, \\ \frac{1}{2}\,(\alpha _j-\alpha _i+1)^2\,\alpha ^r_i &{} &{} \text {if } \alpha _{j}\le \alpha _{i}\le \alpha _{j}+1. \\ \end{array} \right. \end{aligned}$$

To ease the exposition, define auxiliary functions $h_{-}(z)$ and $h_{+}(z)$ corresponding to the two cases of the expected utility function as follows:

$$\begin{aligned} h_{-}(z)&=\left[ 1-\frac{1}{2}\left( z-\alpha _j+1\right) ^2\right] \,z^r,\\ h_{+}(z)&=\frac{1}{2}\,(\alpha _j-z+1)^2\,z^r. \end{aligned}$$

• Case I: $h_{-}(z)$ is maximised on the interval $[\underline{z},\alpha _j]$ at $z^{*}_{-}=\min \{z_1,\,\alpha _j\}$, where $z_1$ is defined further below. The first derivative of $h_{-}(z)$ is

  $$\begin{aligned} h'_{-}(z)=\frac{r\,z^{r-1}}{2}\, \left[ 2-(z-\alpha _j+1)^2-\frac{2z}{r}\,(z-\alpha _j+1)\right] \end{aligned}$$

  and exhibits two non-zero roots $z_1,z_2\ne 0$ such that

  $$\begin{aligned} z_{1,2}=\frac{(1+r)\,(\alpha _j-1)}{2+r}\pm \frac{1}{2+r}\,\sqrt{2r(2+r)+(\alpha _j-1)^2}. \end{aligned}$$

  It is straightforward to show that $\underline{z}<z_1<\alpha _j-1+\sqrt{2}$. There exists $z'\in (\underline{z},z_1)$ such that $h'_{-}(z')>0$; further, $h'_{-}(z_1)=0$, and $h'_{-}(\alpha _j-1+\sqrt{2})<0$. Because $z_2<\underline{z}$, $z_1$ is the unique root of $h'_{-}(z)$ for $z>\underline{z}$ so that, with continuous differentiability of $h_{-}(z)$ for $z>\underline{z}$ and continuity of $h_{-}(z)$ for $z\ge \underline{z}$, $h_{-}(z_1)$ is the maximum on interval $[\underline{z},\alpha _j-1+\sqrt{2}]$. Therefore, $z^{*}_{-}=z_1$ is the maximiser on interval $[\underline{z},\alpha _j]$ for $z_1\le \alpha _j$ and $z^{*}_{-}=\alpha _j$ emerges as the boundary solution for $z_1>\alpha _j$.

• Case II: $h_{+}(z)$ is maximised on $[\alpha _j,\,\alpha _j+1]$ at $z^{*}_{+}=\max \{z_4,\alpha _j\}$ where $z_4$ is defined further below. The first derivative of $h_{+}(z)$ is

  $$\begin{aligned} h'_{+}(z)=\frac{1}{2}\,z^{r-1}\,(\alpha _j-z+1)\,\left[ r\,(\alpha _j-z+1)-2z\right] \end{aligned}$$

  and exhibits two non-zero roots: $z_3=\alpha _j+1$ and $z_4=r(\alpha _j+1)/(2+r)$. Because $h_{+}(z_3)=0$ and $h_{+}(z)>0$ for $z\in [\alpha _j,\,\alpha _j+1)$, $z_3$ identifies a minimum. It is obvious that $0<z_4<\alpha _j+1$. There exists $z'\in (0,z_4)$ such that $h'_{+}(z')>0$; further, $h'_{+}(z_4)=0$, and there exists $z''\in (z_4,\alpha _j+1)$ such that $h'_{+}(z'')<0$. Because $z_4$ is the unique root of $h'_{+}(z)$ for $0<z<\alpha _j+1$, with continuous differentiability of $h_{+}(z)$ for $0<z<\alpha _j+1$ and continuity of $h_{+}(z)$ for $z\ge 0$, $h_{+}(z_4)$ is the maximum on interval $[0,\alpha _j+1]$. For $\alpha _j\le r/2$, $z_4\ge \alpha _j$; hence, $z^{*}_{+}=z_4$ is the maximiser of $h_{+}(z)$ on interval $[\alpha _j,\alpha _j+1]$ for $\alpha _j\le r/2$. Further, $z^{*}_{+}=\alpha _j$ for $\alpha _j>r/2$ because then $z_4<\alpha _j$.

By $h_{-}(\alpha _j)=h_{+}(\alpha _j)$ and, for $\alpha _j>0$, $h'_{-}(\alpha _j)=h'_{+}(\alpha _j)=\alpha _j^{r-1}\,(r-2\alpha _j)/2$, expected utility $\text {EU}_{i}(\alpha _{i})$ is maximised (i) by $z_1$ for $\alpha _j>r/2$, (ii) by $z_1$ and $z_4$ for $\alpha _j=r/2$ (implying $z_1=z_4$), and (iii) by $z_4$ for $0<\alpha _j<r/2$. The comparison of $h_{-}(z_{-}^{*}=0)=0$ and $h_{+}(z_{+}^{*}=z_4)>0$ implies that $\text {EU}_{i}(\alpha _{i})$ is maximised by $z_4$ for $\alpha _j=0$. The best-response function $\alpha _i^{*}(\alpha _j)$ as given by (4) follows immediately.

B. Rational bidders alongside markdown bidders

In Sect. 2.3 we considered the situation of a heterogeneous population. A share $\rho \in [0,1)$ of the population is perfectly rational. The rest, a share $1-\rho$, consists of markdown bidders. In this appendix we derive the equilibrium conditions for this case.

Let $\theta _j\in \{\text {R}, \overline{\text {R}}\}$ denote the rationality type of player j that can be either fully rational, $\theta _j=\text {R}$, or boundedly rational in the sense of markdown bidding, $\theta _j=\overline{\text {R}}$. Then, a fully rational bidder’s prior probability of competing with another fully rational bidder is $\rho$ and that of facing a markdown bidder is $1-\rho$. Assume there is an equilibrium such that the fully rational type bids according to $\gamma (x)$ and the boundedly rational type bids according to ${\bar{\gamma }}(x)$, where both equilibrium bid functions are strictly increasing. The expected utility of a fully rational bidder i facing the competitor j, who is randomly drawn from the population of bidders, is (assuming that bidder j bids according to the proposed equilibrium) given by:

$$\begin{aligned} \text {EU}_i =\;&\rho \cdot \text {Pr}\{b_i=\max \{b_i,\gamma (x_j)\}\,|\,\theta _j=\text {R}\}\cdot u(x_i-b_i) \\&+(1-\rho )\cdot \text {Pr}\{b_i=\max \{b_i,{\bar{\gamma }}(x_j)\}\,|\,\theta _j=\overline{\text {R}}\}\cdot u(x_i-b_i). \end{aligned}$$

Because the assumed equilibrium bid function of the fully rational type is strictly increasing, there exists the inverse $\chi (b):=\gamma ^{-1}(b)$ that maps a fully rational player’s bid b to the corresponding value x. The probability that player i outbids another fully rational bidder follows as $F(\chi (b_i))$. Let G(b) denote the cumulative distribution function of bids submitted by the boundedly rational type so that the probability of player i outbidding this type is $G(b_i)$. By markdown bidding as described by (5) together with the distribution of values F(x), we have $G(b)=b+r/2$ for $b\in {[-r/2,(2-r)/2]}$. Therefore, the maximization problem of fully rational bidder i that competes with bid $b_i$ against an equilibrium bidder of unknown rationality type is given by

$$\begin{aligned} \max _{b_i}\text {EU}_i = \left[ \rho \, F(\chi (b_i))+(1-\rho )\,G(b_i)\right] \cdot (x_i-b_i)^r . \end{aligned}$$

The first-order condition follows as

$$\begin{aligned}&\left[ \rho F'(\chi (b_i))\chi '(b_i)+(1-\rho ) G'(b_i)\right] (x_i-b_i)^r\\&\quad -r\left[ \rho F(\chi (b_i))+(1-\rho )G(b_i)\right] (x_i-b_i)^{r-1}=0. \end{aligned}$$

For a fully rational bidder i, it cannot be beneficial to deviate from the equilibrium strategy in equilibrium; hence, $x_i=\chi (b_i)$. Using this property and substituting for probability densities yields the following differential equation whose solution (with an appropriate initial value to be determined below) is the inverse of the equilibrium bid function of the fully rational type $\chi (b)$:

$$\begin{aligned} \rho \left[ \chi (b_i)-b_i\right] \chi '(b_i)=\left[ (1+r)\rho -1\right] \chi (b_i)+(1+r)(1-\rho )b_i+(1-\rho )\frac{r^2}{2}.\nonumber \\ \end{aligned}$$

(21)

In equilibrium, a rational bidder with the smallest possible value of 0 never wins against another rational bidder but only against boundedly rational bidders. With the distribution of bids submitted by markdown bidders G(b), the optimal bid of rational bidder i with $x_i=0$ follows as¹²

$$\begin{aligned} \gamma (0)=-\frac{r^2}{2(1+r)}. \end{aligned}$$

The initial condition follows as $\chi (-r^2/(2(1+r)))=0$. Because differential equation (21) is non-linear and non-autonomous, an explicit solution is not known in general. We can, however, derive a good numerical approximation. Figure 3 shows the equilibrium bids for different attitudes towards risk r and various population mixes $\rho$.

For the special case of a population with equilibrium markdown bidders only, i.e., $\rho =0$, Eq. (21) simplifies to

$$\begin{aligned} \chi (b_i)= (1+r) b_i + \frac{r^2}{2} \end{aligned}$$

(22)

implying for a rational bidder who optimises against an equilibrium markdown bidder with $\alpha ^*=r/2$ to bid according to

$$\begin{aligned} b(x) = \frac{x_i}{1+r} - \frac{r^2}{2(1+r)} \,. \end{aligned}$$

(23)

With non-equilibrium markdown bidders who shade valuations by arbitrary $\alpha >0$, so that $G(b)=b+\alpha$ for $b\in [-\alpha ,1-\alpha ]$, a rational bidder’s best-response is

$$\begin{aligned} b(x) = \frac{x_i}{1+r} - \frac{r\,\alpha }{1+r} \end{aligned}$$

(24)

conditional on non-extreme markdowns of $\alpha \le r$, which ensures that a rational bidder with the highest possible valuation does not overbid the highest possible bid of a markdown bidder of $1-\alpha$; otherwise, for $\alpha >r$, there emerges a flat portion of a rational bidder’s best-response, where $b(x)=1-\alpha$ for all $x\ge 1+r-\alpha$. For extreme markdowns, $\alpha \ge 1+r$, the rational bidder – whatever the valuation – would prefer to always win the auction. In this case the best response would be a flat bid of $1-\alpha$ (equal to the highest possible bid of a markdown bidder) for all $x\ge 0$.

C. Further estimation results

1.1 C.1. Estimating Eqs. (6)–(10)

To calculate the posterior distribution for Eqs. (6)–(10), we use JAGS 4.0.0 with 8 different chains, each with 5000 burnin steps. For each chain, we take 10000 actual samples. The posterior distributions for each case are therefore based on 80,000 actual samples.¹³ Table 3 provides the effective sample size and potential scale reduction factor (Gelman and Rubin 1992) for $\beta _3$ from Eq. (7).

Table 3 Convergence statistics for $\beta _3$ from Eq. (7)

1.2 C.2. Estimates and convergence for Eqs. (12)–(20)

For each treatment, we calculate posteriors separately, each time using JAGS with 8 different chains, each chain with 5000 burnin steps. For each chain, we take 10000 samples with a thinning parameter of 10. The posterior distributions for each treatment are therefore based on $8\times 10^{5}$ actual samples.¹⁴ Table 4 shows medians of the distribution for $\delta _c$ from Eq. (19). Numbers are given as percentages. 95% credible intervals are given in brackets.

Table 4 Medians of the distribution for $\delta _c$ from Eq. (19)

The left part of Table 5 shows the potential scale reduction factor for $\delta _c$ . The right part shows the effective sample size.

Table 5 psrf and effective sample size for $\delta _c$ from Equation 19

D. Conducting the experiment and instructions

Participants were recruited by email and could register for the experiment on the internet. At the beginning of the experiment, participants drew balls from an urn to determine their allocation to seats. Being seated, participants then obtained written instructions in German. In the following, we give a translation of the instructions.

After answering control questions on the screen, subjects entered the treatment described in the instructions. After completing the treatment, they answered a short questionnaire on the screen and, then, were paid in cash. The experiment was conducted with z-Tree, (Fischbacher (2007)).

1.1 D.1. General information

You are participating in a scientific experiment that is sponsored by the Deutsche Forschungsgemeinschaft (German Research Foundation). If you read the following instructions carefully, then you can—depending on your decision—gain a considerable amount of money. It is, hence, very important that you read the instructions carefully.

The instructions that you have received are only for your private information. During the experiment, no communication is permitted. Whenever you have questions, please raise your hand. We then answer your question at your seat. Not following this rule leads to exclusion from the experiment and all payments.

During the experiment, we do not talk about Euro, but about ECU (Experimental Currency Unit). Your entire income is first determined in ECU. The total amount of ECU that you have obtained during the experiment is converted into Euro at the end and paid to you in cash. The conversion rate is shown on your screen at the beginning of the experiment.

1.2 D.2. Information regarding the experiment

Today you are participating in an experiment on auctions. The experiment is divided into separate rounds. We conduct 12 rounds. In the following, we explain what happens in each round.

In each round, you bid for an object that is auctioned. Together with you, another participant also bids for the same object. Hence, in each round, there are two bidders. In each round, you are allocated randomly to another participant for the auction. Your co-bidder in the auction changes in every round. The bidder with the highest bid obtains the object. If bids are the same, the object is allocated randomly.

For the auctioned object you have a valuation in ECU. This valuation is between x and $x+50$ ECU and is determined randomly in each round.¹⁵ The range from x to $x+50$ is shown to you at the beginning of the experiment on the screen and is the same in each round.¹⁶From this range you obtain new and random valuations for the object in each round. The other bidder in the auction also has a valuation for the object. The valuation that the other bidder attributes to the object is determined by the same rules as your valuation and changes in each round, too. All possible valuations of the other bidder are also in the interval from x to $x+50$ from which also your valuations are drawn. All valuations between x and $x+50$ are equally probable. Your valuations and those of the other player are determined independently. You will be told your valuation in each round. You will not know the valuation of the other bidder.

1.2.1 D.2.1. Experimental procedure

The experimental procedure is the same in each round and is described in the following. Each round in the experiment has two stages.

1. Stage

In the first stage of the experiment, you see the following screen:¹⁷

At that stage you do not know your own valuation for the object in this round. On the right side of the screen, you are asked to enter a bid for six hypothetical valuations that you may have for the object. These six hypothetical valuations are x, $x+10$, $x+20$, $x+30$, $x+40$, and $x+50$ ECU. Your input into this table will be shown in the graph on the left side of the screen when you click on “draw bids”. In the graph, the hypothetical valuation is shown on the horizontal axis, the bids are shown on the vertical axis. Your input in the table is shown as six points in the diagram. Neighbouring points are connected with a line automatically. These lines determine your bid for all valuations between the six points for those you have made an input. For the other bidder, the screen in the first stage looks the same and there are as well bids for six hypothetical valuations. The other bidder cannot see your input.

2. Stage The actual auction takes place in the second stage of each round. In each round, we play not only a single auction but five auctions. This is done as follows: Five times a random valuation is determined that you have for the object. Similarly for the other bidder five random valuations are determined. You see the following screen:¹⁸

For each of your five valuations, the computer determines your bid according to the graph from stage 1. If a valuation is precisely at x, $x+10$, $x+20$, $x+30$, $x+40$, or $x+50$ the computer takes the bid that you gave for this valuation. If a valuation is between these points, your bid is determined according to the joining line. In the same way, the bids of the other bidder are determined for his five valuations. Your bid is compared with the one of the other bidder. The bidder with the higher bid obtains the object.

Your income from the auction: For each of the five auctions the following holds:

• The bidder with the higher bid obtains the valuation he had for the object in this auction added to his account minus his bid for the object.

• If the bidder with the higher bid has a negative valuation for the object, the ECU account is reduced by this amount.¹⁹

• If the bid of bidder with the higher is a negative number, the amount is added to his ECU account.²⁰

• The bidder with the smaller bid obtains no income from this auction.

You total income in a round is the sum of the ECU income from those auctions in this round where you have made the higher bid.

This ends one round of the experiment and you see the input screen from stage 1 again in the next round.

At the end of the experiment, your total ECU income from all rounds will be converted into Euro and paid to you in cash together with your Show-Up Fee of 3.00 Euro.

Please raise your hand if you have questions.