An experiment on auctions with endogenous budget constraints

Abstract

We perform laboratory experiments comparing auctions with endogenous budget constraints. A principal imposes a budget limit on a bidder (an agent) in response to a principal-agent problem. In contrast to the existing literature where budget constraints are exogenous, this theory predicts that tighter constraints will be imposed in first-price auctions than in second-price auctions, tending to offset any advantages attributable to the lower bidding strategy of the first-price auction. Our experimental findings support this theory: principals are found to set significantly lower budgets in first-price auctions. The result holds robustly, whether the principal chooses a budget for human bidders or computerized bidders. We further show that the empirical revenue difference between first- and second-price formats persists with and without budget constraints.

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Notes

  1. 1.

    See Cox et al. (1982, 1988) as the seminal papers, and Kagel (1995) for a detailed survey. Risk aversion (Cox et al. 1988), anticipation of regret (Filiz-Ozbay and Ozbay 2007), joy of winning (see, for example, Goeree et al. 2002), fear of losing (Delgado et al. 2008; Cramton et al. 2012a, b), and level-k thinking (Crawford and Iriberri 2007) have been offered as possible explanations of the overbidding phenomenon.

  2. 2.

    One interpretation of the results from Burkett (2015) is that the budgets in the model function like bids that are not always “active”. If the subjects recognize this, one might expect similarities between the budgeting decisions in this experiment and bidding decisions in the existing literature.

  3. 3.

    One could imagine other mechanisms for constraining the bidding behavior of the agent. Burkett (2016) shows that the current method is optimal in a general sense if the agent is protected by limited liability and the conditional distribution of the agent’s signal satisfies certain assumptions, which are satisfied in the special case used here.

  4. 4.

    As will be clear from our equilibrium analysis, the principal’s signal is irrelevant information for the bidder in this setup since the equilibrium unconstrained bid is a function of only the bidder’s valuation.

  5. 5.

    The payoffs are proportional to those expressions to avoid double counting the total profits. For example, the bidder and the principal might be equity holders in a firm with shares \(\sigma_{b}\) and \(\sigma_{p}\), respectively (where \(\sigma_{b} + \sigma_{p} \le 1\)). The bidder is assumed to receive \(\sigma_{b} \left( {t_{i} - p } \right)\) and the principal to receive \(\sigma_{p} \left( {\delta t_{i} - p } \right)\). This formulation identifies the term \(\left( {1 - \delta } \right)\sigma_{b} t_{i}\) (the difference between the bidder’s payoff and \(\sigma_{b} \left( {\delta t_{i} - p } \right)\)) as the bidder’s private payoff from obtaining the good.

  6. 6.

    This assumes that \(w\left( s \right)\) lies in the range of \(b\left( t \right)\), but this must be true in equilibrium (see Burkett 2015).

  7. 7.

    Proposition 2 in the “Appendix” states the uniqueness property of this equilibrium.

  8. 8.

    Although \(b_{FP} \left( . \right)\) Eq. (2) looks complicated it is approximately linear for the δ used in our experiments (see Fig. 4).

  9. 9.

    In fact, one can make the stronger assertion that the two auction formats agree in their allocations for every possible realization of the signals. This is a consequence of the winner being the one with the highest value of \({ \hbox{min} }\left\{ {t,\hat{t}\left( s \right)} \right\}\) in both cases.

  10. 10.

    EEL-UMD is a relatively new lab and one or two auction experiments are conducted in a year. So we are confident that our very rich subject pool is not overly experienced in auction experiments.

  11. 11.

    We thank the editors for recommending this control treatment to see whether the revenue gap between different auction formats is getting larger or not with the introduction of budget constraints.

  12. 12.

    The random draws were balanced within the active principal treatments not in between. This is because in the main treatments, we had eight bidders and eight principals in a session and in the control treatments we had sixteen principals in the lab where the bidders were computerized players.

  13. 13.

    In a typical session, the instructions were described for 20–30 min while the actual play lasted for about an hour.

  14. 14.

    In the experiment, we referred to each principal as Participant A and each agent as Participant B, to avoid any name driven bias.

  15. 15.

    They learned the opponent’s payoff when the opponent lost—it must have been zero—but we did not tell them the opponent’s payoff when the opponent wins because, in that case, the subjects could determine the actual value of the opponent and his bidding strategy to some extent. Since we used random matching in each round to generate single-shot games, we aimed to minimize the learning about the strategy of the other subjects.

  16. 16.

    We set \(\delta = 2/5\) in the experiments because for this value of \(\delta\), the equilibrium strategies of first price auction are approximately linear.

  17. 17.

    We are aware that if the principals do not play optimally against such computerized bidders, we will not see equilibrium plays since the computers cannot respond to principals’ strategies. However, this design will still allow us to compare the budget decisions of the principals across different auctions and whether the difference in budgets is in the same direction as the theory predicts. Moreover, since we know the bidders’ strategies, we can compute what types will be restricted by each budget set.

  18. 18.

    Note that when the principal is passive and cannot set bid cap for her bidder, the bidder who values the auctioned item 2.5 times more than the principal may cause the principal to lose a lot of money.

  19. 19.

    Bankruptcy is always a potential problem in auction experiments. We assured our subjects that they will earn positive amounts.

  20. 20.

    We are confident that using different exchange rates does not alter our findings since our findings in the main treatments and in the control treatments (where the agents are computerized and therefore there is only principals’ exchange rate) are qualitatively the same.

  21. 21.

    An alternative method to balance the earnings of principals and bidders could be to provide them with different endowments. We did not use this method since we wanted to keep the relative weights of the variable and fixed portions of the bidders’ expected payoff comparable for different roles.

  22. 22.

    We also performed t-tests by using each observation and the results were not qualitatively different in any of the comparisons except for the revenues in SP experiments and SP equilibrium prediction for computerized bidders in Table 7.

  23. 23.

    In the analysis of the treatments with passive principals we treat each session as two independent sessions run in parallel. These sessions were structured as two parallel sub-sessions in which each set of eight bidders were only matched to other bidders in the same set. The bidder value draws in each sub-session correspond to one session from the human bidder treatments.

  24. 24.

    Revenues were not significantly different between the treatments with human bidders and those with computerized bidders for both the first-price auction (p = 0.690) and the second-price auction (p = 0.690).

  25. 25.

    The argument that a risk-averse principal should reduce her budget in the second-price auction is robust in the sense that it only depends on her bidder using the weakly dominant strategy of bidding the minimum of his value and the budget, which would be the optimal choice for the bidder regardless of whether he is assumed to be risk averse or risk neutral. The argument is also independent of the specified preferences of the opposing principals and bidders.

  26. 26.

    The only significant difference is between the fraction of realized surplus in the SP auction without budget constraints and the fraction of realized surplus in the SP auction with computerized bidders (p = 0.032), but this effect was in the opposite direction (surplus fell without budget constraints).

  27. 27.

    One reason for the tendency for overbidding in second-price could be the left-skewed value distributions. The literature argued that the subjects have difficulty learning not to overbid in second-price because they are rarely confronted with the consequences of their “mistake” (see Kagel and Levin 1993; Cooper and Fang 2008; Garratt et al. 2012).

  28. 28.

    The revenue gap measures we used were the difference between first- and second-price revenue and the ratio of second-price to first-price revenue. We use first-price revenue in the denominator of the latter because we observed near-zero revenue in several of the second-price auctions. For each measure we tested that the session averages for the human bidder with and without budget treatments were different using a MWW test (p > 0.413). We also used the MWW test on these measures using the disaggregated individual auction data, which we matched across treatments to use the same draws (400 auctions per treatment) and found no significant results at the 5% level (p > 0.267).

  29. 29.

    The box plots were created using standard techniques. The white lines represent the median; the box represents the interquartile range (IQR); the whiskers extend to the furthest data point within 1.5 × IQR; and the open circles are individual data points outside 1.5 × IQR. In Fig. 1, 24 out of 28 of the outliers in the second-price auction represent decisions made by one subject.

  30. 30.

    Note that the data from the second price auctions with computerized bidders is noisier than its counterpart with human bidders. With human bidders only 28 of 1000 budget decisions were above 100 and with computerized bidders 179 of 2000 observations were above 100 in Figs. 1 and 3, respectively.

  31. 31.

    In a separate analysis, we calculated the \(R^{2}\) values from regressions of the budget on the principal signals for each individual principal. For principals in the first-price auction, 75% of the principals had \(R^{2}\) values above 0.79, 50% were above 0.87 and 25% were above 0.93. The corresponding numbers in the second-price auction were 0.87, 0.94 and 0.97. With computerized bidders in the first-price auction (second-price auction), 75% of the principals had \(R^{2}\) values above 0.72 (0.79), 50% were above 0.86 (0.92), and 25% were above 0.93 (0.96).

  32. 32.

    We controlled for the round number by including dummy variables indicating the first 10 rounds of the experiment in each treatment. However, the dummy variables were significant only for the first-price auction and did not affect the estimates of interest when they were included, so they are excluded here.

  33. 33.

    With human bidders, the \(\chi^{2}\) test statistic for \(\beta_{2} = 0.625\) is 83.94 \((p = 0.000)\) and for \(\beta_{2} + \beta_{3} = 0.276\) it is 145.04 \((p = 0.000)\). With computerized bidders, the \(\chi^{2}\) test statistic for \(\beta_{2} = 0.625\) is 557.34 \((p = 0.000)\) and for \(\beta_{2} + \beta_{3} = 0.276\) it is 96.67 \((p = 0.000)\).

  34. 34.

    Note that we continue to reject this hypothesis if we exclude the first session of the second-price treatment. The subject who set the outlier budget levels in Fig. 1 participated in that session.

  35. 35.

    The larger fraction of outliers evident in the second-price treatments might be the result of the noisier feedback from the second-price design. The negative consequence of setting a high budget in either treatment is that one might have to (possibly) pay too high of a price for the item. In the first-price auction, the realization of this consequence requires that one’s bidder also place a high bid, but in the second-price auction one’s bidder must place a high bid and one’s opponent must have a high budget and place a high bid which is rare.

  36. 36.

    Instead of the vertical line at the mean value, if we insert the density of equilibrium budget/signal for FP in these Figures, we need to draw a density function with very small variance such that it concentrates around its mean (0.276) and its peak is too high to include in these Figures. That’s why we present just the vertical line passing through the mean of equilibrium budget/signal realizations here.

  37. 37.

    Choices \(w < 0\) win with zero probability given the description of the opponents’ behavior and are weakly dominated by \(w^{{\prime }} = 0\). Similarly, any \(w > 1\) leaves the bidder unconstrained with probability one and is weakly dominated by \(w^{{\prime }} = 1\).

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Acknowledgements

We thank the National Science Foundation (Grant SES-09-24773) for its support. We are grateful to our colleagues, Erkut Ozbay and Peter Cramton, for valuable discussions, to Kristian Lopez-Vargas for his assistance in programming the experiment, and to seminar participants at Carnegie Mellon-Tepper School of Business, the ESEI Market Design Conference at CERGE-EI, and the 2012 North-American ESA Conference for their helpful comments.

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Correspondence to Emel Filiz-Ozbay.

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Appendix

Appendix

Derivation of equilibrium

Let \(t\) represent the valuation of the bidder, \(\delta t\) represent the valuation of the principal, and \(s\) represent the signal received by the principal. The distributions of these random variables in the experiment are:

$$s \sim U\left[ {0,100} \right] ;$$
$$t \sim U\left[ {0,\frac{s}{\delta }} \right] ;{\text{and}}$$
$$\delta t \sim U\left[ {0,s} \right] .$$

The relevant density functions are:

$$f\left( s \right) = \frac{1}{100} , 0 \le s \le 100 ;{\text{and}}$$
$$f(t|s) = \frac{\delta }{s},\quad 0 \le t \le \frac{\delta }{s}.$$

Second-price auction

Suppose that opposing bids take the form \(\hbox{min} \left\{ {t_{j} ,\frac{{s_{j} }}{2 - \delta }} \right\}\), and that the density of the opposing bids is given by \(g_{SP} \left( x \right)\). Given any budget \(w\), it is a weakly dominant strategy in this environment for bidder \(i\) to bid \(\hbox{min} \left\{ {t_{i} ,w} \right\}\). Now consider principal \(i\)’s choice of budget, which is equivalent to a choice of \(\hat{t}_{SP} \left( s \right) \equiv w_{SP} \left( s \right)\). Principal \(i\)’s objective is

$$\mathop \int \limits_{0}^{{\hat{t}_{SP} }} \mathop \int \limits_{0}^{t} \left( {\delta t - x} \right)g_{SP} \left( x \right)f (t | s) dx dt + \mathop \int \limits_{{\hat{t}_{SP} }}^{{\frac{s}{\delta }}} \mathop \int \limits_{0}^{{\hat{t}_{SP} }} \left( {\delta t - x} \right)g_{SP} \left( x \right)f (t | s) dx dt .$$

The first term represents the payoff in the event that the budget constraint does not bind, and the second is the payoff when the constraint does bind. The first order condition is:

$$g_{SP} \left( {\hat{t}_{SP} } \right)\mathop \int \limits_{{\hat{t}_{SP} }}^{s/\delta } \left( {\delta t - \hat{t}_{SP} } \right) f (t | s) dt = 0$$
$$\mathop \int \limits_{{\hat{t}_{SP} }}^{s/\delta } \left( {\delta t - \hat{t}_{SP} } \right)\frac{\delta }{s}dt = 0$$
$$\frac{s}{{s - \delta \hat{t}_{SP} }}\mathop \int \limits_{{\hat{t}_{SP} }}^{s/\delta } \left( {\delta t - \hat{t}_{SP} } \right)\frac{\delta }{s}dt = 0$$
$$E\left( {\delta t - \hat{t}_{SP} |t > \hat{t}_{SP} ,s} \right) = 0$$
$$\frac{{s + \delta \hat{t}_{SP} }}{2} - \hat{t}_{SP} = 0$$
$$\hat{t}_{SP} = \frac{s}{2 - \delta } .$$

For the second line, note that \(g_{SP} \left( {\hat{t}_{SP} } \right)\) is a constant. Note that this implies that the principal’s optimal choice of budget does not depend on the distribution of opposing bids. In the third line, we are dividing by \(P (t > \hat{t}_{SP} | s)\) to make the left side a conditional expectation. The final line is the principal’s equilibrium choice of \(\hat{t}_{SP}\), which in the second-price auction is also the equilibrium choice of budget constraint. To verify that this choice of \(\hat{t}_{SP}\) maximizes the principal’s objective, notice that the sign of the principal’s first order condition is negative for \(\hat{t} > \hat{t}_{SP}\) and positive for \(\hat{t} < \hat{t}_{SP}\). Finally, it is easy to verify that any choice for \(w\) such that \(w \notin \left[ {0,1} \right]\) is weakly dominated by some choice \(w' \in \left[ {0,1} \right]\).Footnote 37 Therefore, in equilibrium all bids take the form \(\hbox{min} \left\{ {t,\frac{s}{2 - \delta }} \right\}\).

First-price auction

Suppose that a type \(t\) bidder in a first-price auction wins with probability \(G_{FP} \left( t \right)\) (assume for the moment that \(G_{FP} \left( t \right)\) is differentiable a.e. and increasing and let \(g_{FP} \left( t \right)\) be the corresponding density function), then a standard analysis concludes that the optimal choice of bid is given by

$$b_{FP} \left( t \right) = \frac{1}{{G_{FP} \left( t \right)}}\mathop \int \limits_{0}^{t} x g_{FP} \left( x \right)dx$$

when it is feasible. To analyze the constrained bidder’s choice, observe that \(b_{FP} \left( t \right)\) is nondecreasing and continuous, so a budget constraint \(w_{FP} \left( s \right)\) can be equivalently represented as a cutoff type, \(\hat{t}_{FP} \left( s \right)\) defined by \(w_{FP} \left( s \right) = b_{FP} \left( {\hat{t}_{FP} \left( s \right)} \right)\). Write the bidder’s objective in terms of a choice of type \(t^{{\prime }}\) as

$$\pi_{B} \left( {t,t^{{\prime }} } \right) = G_{FP} \left( {t^{{\prime }} } \right)\left( {t - b_{FP} \left( {t^{{\prime }} } \right)} \right) = G_{FP} \left( {t^{{\prime }} } \right)\left( {t - t^{{\prime }} } \right) + \mathop \int \limits_{0}^{{t^{{\prime }} }} G_{FP} \left( x \right)dx,$$

where the equality follows after integrating by parts. The derivative of this expression with respect to \(t^{{\prime }}\) is negative for \(t^{{\prime }} > t\) and positive for \(t^{{\prime }} < t\). It follows that a bidder who is restricted to setting \(t^{{\prime }} \le \hat{t}_{FP}\) optimally sets \(t^{{\prime }} = t\) if \(t \le \hat{t}_{FP}\) and \(t^{{\prime }} = \hat{t}_{FP} \left( s \right)\), otherwise.

The bidder’s behavior allows us to write the principal’s objective in terms of the choice of \(\hat{t}_{FP}\) as

$$\mathop \int \limits_{0}^{{\hat{t}_{FP} }} G_{FP} \left( t \right)\left( {\delta t - b_{FP} \left( t \right)} \right)f (t | s)dt + \mathop \int \limits_{{\hat{t}_{FP} }}^{s/\delta } G_{FP} \left( {\hat{t}_{FP} } \right)\left( {\delta t - b_{FP} \left( {\hat{t}_{FP} } \right)} \right) f (t | s) dt .$$

Pugging in for \(b_{FP} \left( t \right)\) this becomes:

$$\mathop \int \limits_{0}^{{\hat{t}_{FP} }} \left( {G_{FP} \left( t \right)\delta t - \mathop \int \limits_{0}^{t} x g_{FP} \left( x \right)dx} \right)f (t | s)dt + \mathop \int \limits_{{\hat{t}_{FP} }}^{s/\delta } \left( {G_{FP} \left( {\hat{t}_{FP} } \right)\delta t - \mathop \int \limits_{0}^{{\hat{t}_{FP} }} x g_{FP} \left( x \right)dx} \right) f (t | s) dt .$$

As with the second price auction, the solution for \(\hat{t}_{FP}\) does not depend on \(G_{FP} \left( t \right)\) and is actually the same as \(\hat{t}_{SP}\):

$$\hat{t}_{FP} = \frac{s}{2 - \delta } .$$

The logic of Footnote 33 holds in the first-price case as well. Summarizing, if the probability of a type \(t\) bidder winning is \(G_{FP} \left( t \right)\), then the bidder optimally bids \(b_{FP} \left( {\hbox{min} \left\{ {t,\frac{s}{2 - \delta }} \right\}} \right)\). In a symmetric equilibrium, the bid functions are the same for each bidder and the winner is the bidder with the higher value of \(\hbox{min} \left\{ {t,\frac{s}{2 - \delta }} \right\}\). The distribution of \(\hbox{min} \left\{ {t,\frac{s}{2 - \delta }} \right\}\) determines \(G_{FP} \left( t \right)\). We find

$$G_{FP} \left( t \right) = \left\{ {\begin{array}{*{20}l} {\left[ {2 - \delta - \delta \ln \left( {\left( {\frac{2 - \delta }{100}} \right)t} \right)} \right]\frac{t}{100},} \hfill & {t \in \left[ {0,\frac{100}{2 - \delta }} \right]} \hfill \\ {1,} \hfill & {t > \frac{100}{2 - \delta }} \hfill \\ \end{array} } \right.$$

This function is differentiable a.e. and increasing as was originally assumed. This makes the unconstrained bid function

$$b_{FP} \left( t \right) = \left[ {\frac{{4 - 3\delta - 2\delta \ln \left( {\frac{2 - \delta }{100}t} \right)}}{{4 - 2\delta - 2\delta \ln \left( {\frac{2 - \delta }{100}t} \right)}}} \right]\frac{t}{2}, t \in \left[ {0,\frac{100}{2 - \delta }} \right].$$

Proof of proposition 1

Suppose the principals choose a cutoff type strategy according to \(\hat{t}_{\alpha } \left( s \right) = \alpha s\), where in equilibrium \(\alpha = \alpha^{*} = 5/8\) in both auction formats, and let the distribution of \(\hat{t}_{\alpha } \equiv \hbox{min} \left\{ {t,\alpha s} \right\}\) be given by \(G_{\alpha } \left( x \right)\) with \(G_{\alpha }^{\left( i \right)} \left( x \right)\) being the distribution of the \(ith\) order statistic. Finally, denote the equilibrium unconstrained bid function derived above by \(b_{{\alpha^{*} }} \left( x \right) = \frac{1}{{G_{{\alpha^{*} }} \left( x \right)}}\mathop \int \limits_{0}^{x} y dG_{{\alpha^{*} }} \left( y \right)\). Then the experimental expected revenue in the first-price auction \(\left( {E\left[ {R_{\alpha }^{FP} } \right]} \right)\) and the second-price auction \(\left( {E\left[ {R_{\alpha }^{SP} } \right]} \right)\) with principals following a strategy \(\hat{t}_{\alpha } \left( s \right)\) and the computerized bidders following the corresponding equilibrium unconstrained bid functions can be written as:

$$E\left[ {R_{\alpha }^{FP} } \right] = \mathop \int \limits_{0}^{100\alpha } b_{{\alpha^{*} }} \left( x \right) dG_{a}^{\left( 1 \right)} \left( x \right)\quad {\text{and}}\quad E\left[ {R_{\alpha }^{SP} } \right] = \mathop \int \limits_{0}^{100\alpha } x dG_{a}^{\left( 2 \right)} \left( x \right).$$

In the formulas above, note that the integral limits are determined by the strategy of the principal, i.e. \(\hat{t}_{\alpha } \left( s \right) = \alpha s\) and the integrands are determined by the bid functions of the computerized bidders who follow the equilibrium unconstrained bid functions, i.e. \(b_{{\alpha^{*} }} \left( x \right)\) in FP and value bidding in SP. As the theory shows, with \(\alpha = \alpha^{*}\) the two expressions are equal. Also, using a standard revenue equivalence argument we have:

$$E\left[ {R_{\alpha }^{SP} } \right] = \mathop \int \limits_{0}^{100\alpha } b_{\alpha } \left( x \right) dG_{a}^{\left( 1 \right)} \left( x \right) ,$$

where \(b_{\alpha } \left( x \right)\) is defined analogously to \(b_{{\alpha^{*} }} \left( x \right)\). So the first-price auction raises more revenue when \(\alpha > \alpha^{*}\) if and only if:

$$\begin{aligned} & \mathop \int \limits_{0}^{100\alpha } b_{{\alpha^{*} }} \left( x \right) dG_{a}^{\left( 1 \right)} \left( x \right) > \mathop \int \limits_{0}^{100\alpha } b_{\alpha } \left( x \right) dG_{a}^{\left( 1 \right)} \left( x \right) \\ & \quad \Leftrightarrow \mathop \int \limits_{0}^{100\alpha } \left( {b_{{\alpha^{*} }} \left( x \right) - b_{\alpha } \left( x \right)} \right) dG_{a}^{\left( 1 \right)} \left( x \right) > 0 . \\ \end{aligned}$$

In fact, for \(\alpha > \alpha^{*}\) and all \(x > 0\), \(b_{{\alpha^{*} }} \left( x \right) > b_{\alpha } \left( x \right)\), so that the above expression holds. We calculate:

$$b_{\alpha } \left( x \right) = \left( {1 - \frac{\alpha }{{5 - 2\alpha \ln \left( {\frac{\text{x}}{100\alpha }} \right)}}} \right)\frac{x}{2} .$$

So:

$$b_{{\alpha^{*} }} \left( x \right) = \left( {1 - \frac{1}{{8 - 2\ln \left( {\frac{2x}{125}} \right)}}} \right)\frac{x}{2} .$$

Therefore for \(\alpha \in \left( {\frac{5}{8},\frac{5}{2}} \right]\):

$$\begin{aligned} & b_{{\alpha^{*} }} \left( x \right) - b_{\alpha } \left( x \right) > 0 \\ & \quad \Leftrightarrow \frac{\alpha }{{5 - 2\alpha \ln \left( {\frac{x}{100\alpha }} \right)}} > \frac{1}{{8 - 2\ln \left( {\frac{2x}{125}} \right)}} \\ & \quad \Leftrightarrow 8\alpha - 5 > 2\alpha \ln \left( {\frac{8\alpha }{5}} \right) , \\ \end{aligned}$$

which holds for \(\alpha\) in this range. Note that for \(\alpha > \frac{5}{2}\) the bidder is never constrained.

Proposition 2

The equilibrium in the first-price auction with budget constraints is the unique equilibrium with increasing and differentiable budget functions.

Proof of Proposition 2

Suppose that there exists another (possibly asymmetric) equilibrium in this model with the property that the budget functions \((w_{i} \left( {s_{i} } \right))\) are increasing and differentiable in the principals’ signals. Each pair of budget functions leads to a unique equilibrium being played between the bidders. This follows from two results in Maskin and Riley (2003). First, their Lemma 2 shows that bidders best response functions are nondecreasing in the range of potentially winning bids, which implies that budget functions can be equivalently thought of in terms of cutoff types \((\hat{t}_{i} (s_{i} ))\). We can then think of the auction as occurring between two bidders with types \(\hbox{min} \left\{ {t_{i} ,\hat{t}_{i} \left( {s_{i} } \right)} \right\},\) i = 1, 2. Proposition 1 in Maskin and Riley (2003) then implies that the equilibrium of this auction is unique. So if the principals both used the budgets prescribed in the equilibrium derived in this paper there is only one bidding equilibrium in the auction game.

Therefore, if there exists another equilibrium with increasing, differentiable budget functions, both budget functions must differ from the one derived above. Suppose that in such an equilibrium a type \(t\) bidder’s probability of winning is \(G_{FP}^{*} \left( t \right)\). The argument in the previous section implies that in this case the principal’s optimal choice of cutoff type is \(\frac{s}{2 - \delta }\), which is a contradiction. The critical observation is that the principal’s optimal choice of cutoff type does not depend on the equilibrium being played in the subsequent auction.

Proposition 3

A risk-averse principal sets a lower budget in the second-price auction than the risk-neutral principal does when her bidder uses the weakly dominant strategy of bidding the minimum of the realized value and the budget.

Proof of proposition 3

Let \(u_{i} \left( x \right)\) be the increasing, concave Bernoulli utility function of principal \(i\) where \(x\) is the ex post monetary payoff. If the distribution of opposing bids is \(G\left( x \right)\) and bidder \(i\) bids her value, principal \(i\) payoff is

$$\mathop \int \limits_{0}^{{\hat{t}}} \left( {\mathop \int \limits_{0}^{t} u_{i} \left( {\delta t - x} \right)g\left( x \right)dx + \left( {1 - G\left( t \right)} \right)u_{i} \left( 0 \right)} \right)\frac{\delta }{s}dt + \mathop \int \limits_{{\hat{t}}}^{s/\delta } \left( {\mathop \int \limits_{0}^{{\hat{t}}} u_{i} \left( {\delta t - x} \right)g\left( x \right)dx + \left( {1 - G\left( {\hat{t}} \right)} \right)u_{i} \left( 0 \right)} \right)\frac{\delta }{s}dt.$$

The derivative with respect to \(\hat{t}\) has the same sign as

$$E\left[ {u_{i} \left( {\delta t - \hat{t}} \right) |t \ge \hat{t},s} \right] - u_{i} \left( 0 \right).$$

Due to risk aversion

$$E\left[ {u_{i} \left( {\delta t - \hat{t}} \right) |t \ge \hat{t},s} \right] - u_{i} \left( 0 \right) < u_{i} \left( {E\left[ {\delta t - \hat{t} |t \ge \hat{t},s} \right]} \right) - u_{i} \left( 0 \right),$$

where the second term has the same sign as the derivative of the risk-neutral principal’s objective. Therefore, the marginal payoff to raising \(\hat{t}\) is lower under risk aversion and as a result the choice of \(\hat{t}\) must be smaller. Note that this is effectively a single agent decision problem, since G does not affect the sign of the marginal payoff.

Proposition 4

Modify the model so that principals’ monetary payoff decreases by the loser regret term of \({ \hbox{max} }\left( {\alpha \left( {value - price} \right),0} \right)\) with \(\alpha > 0\) and bidders’ payoff decreases by the loser regret term of \({ \hbox{max} }\left( {\alpha \left( {\hbox{min} \left( {value,budget} \right) - price} \right),0} \right)\), in the event that they lose the auction. Then the equilibrium budgets are higher in the second-price auction than the no regret-case.

Proof of Proposition 4

First, it remains an equilibrium for the bidders to bid \({ \hbox{min} }\left( {value,budget} \right)\), because they never experience loser regret and this is optimal without loser regret. Letting \(G\left( x \right)\) be the distribution of opposing bids, when \(\hat{t} < s\) the principal’s payoff is now

$$\mathop \int \limits_{0}^{{\hat{t}}} \left( {\mathop \int \limits_{0}^{t} \left( {\delta t - x} \right)g\left( x \right)dx} \right)\frac{\delta }{s}dt + \mathop \int \limits_{{\hat{t}}}^{{\hat{t}/\delta }} \left( {\mathop \int \limits_{0}^{{\hat{t}}} \left( {\delta t - x} \right)g\left( x \right)dx} \right)\frac{\delta }{s}dt + \mathop \int \limits_{{\hat{t}/\delta }}^{s/\delta } \left( {\mathop \int \limits_{0}^{{\hat{t}}} \left( {\delta t - x} \right)g\left( x \right)dx - \alpha \mathop \int \limits_{{\hat{t}}}^{\delta t} \left( {\delta t - x} \right)g\left( x \right)dx} \right)\frac{\delta }{s}dt.$$

Note that it is only possible for the principal to experience loser regret if \(\hat{t}/\delta < t\). Differentiating with respect to \(\hat{t}\) we get the expression

$$\frac{{g\left( {\hat{t}} \right)s}}{\delta }\left( {\mathop \int \limits_{{\hat{t}}}^{{\frac{{\hat{t}}}{\delta }}} \left( {\delta t - \hat{t}} \right)dt + \mathop \int \limits_{{\frac{{\hat{t}}}{\delta }}}^{{\frac{s}{\delta }}} \left( {1 + \alpha } \right)\left( {\delta t - \hat{t}} \right)dt} \right).$$

Therefore, the derivative with respect to \(\hat{t}\) shifts upwards for all \(\hat{t} < s\), implying that the choice of \(\hat{t}\) must weakly increase in \(\alpha\) or all \(\hat{t} < s\). Recall that using the parameters from the experiment \(\hat{t} = \left( {5/8} \right)s\).

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Ausubel, L.M., Burkett, J.E. & Filiz-Ozbay, E. An experiment on auctions with endogenous budget constraints. Exp Econ 20, 973–1006 (2017). https://doi.org/10.1007/s10683-017-9520-9

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Keywords

  • Auctions
  • Endogenous budget constraint
  • Principal-agent problem

JEL Classification

  • D44
  • C91