Pay–per–bid Auctions Hinnerk Gnutzmann 10 April 2011 - Preliminary & Incomplete In pay–per–bid auctions, bidders pay a fee for each bid they place – which is sunk, irrespective of whether the auction is eventually won or lost. Two such auctions that have recently become popular on the Internet – the penny auction and the lowest unmatched bid auction. In both cases, the seller makes in expectation much higher revenue than the value of the good being sold. We develop a simple model in which bidders estimate the probability of their bidding winning with some (small) noise, and show that this noise may be amplified by the pay–per–bid auction to yield large excess profit for the seller.

Contents 1 Introduction

2

2 Exploring Auctories 2.1 Penny Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lowest Unmatched Bid Auction . . . . . . . . . . . . . . . . . . . . . . .

3 3 4

3 Model 3.1 Structure of the Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Payoff Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 8

4 Adding Behavioural Bidders 4.1 Entertainment Value of Auctions 4.1.1 Joy of Winning . . . . . . 4.2 Optimists and Pessimists . . . . 4.3 Revenue Comparisions . . . . . .

. . . .

5 Conclusion

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

10 10 11 11 13 14

1

1 Introduction The market design of eBay–type online auctions is already well studied (e.g. Ockenfels and Roth [2006],Ockenfels et al. [2006]), but recently new selling mechanisms have appeared on the Internet. These pay–per–bid auctions require a fee to be paid for each bid that is placed. The fee is sunk, irrespective of whether the bidder ends up winning or losing the auction. For example, a penny auction can be understood as an ascending auction with a bidding fee. Thus it may cost a bidder 50p to place a bid that raises the current price by 10p. If she is not overbid, she wins the good at the current price – otherwise the fee is lost. In a sense, placing a bid here can be seen as buying a lottery ticket. But the odds of winning are determined endogenously by the probability that other bidders will place a higher bid. In a lowest unmatched bid auction, the participating bidders simultaneously submit a sealed bid. The agent with the lowest bid that was not placed by any other bidder wins and pays his bid; all bidders pay the bidding fee. Variations in which the highest unique bid wins are also used on the Internet. Equilibrium strategies in this framework are complex and have so far been computed only numerically in special cases Rapoport et al. [2009]; from the bidders perspective, the auction must seem mostly like a game of chance. Because these pay–per–bid auctions combine elements of auctions and lotteries, we shall refer to them as “auctories” in this paper. In this paper, we consider two explanations for the emergence and apparent success of these auctories. First, the winning bidder in general pays much less than the value of the object; when bidders are budget–constrained, such auctories can break the indivisibility of a single high–value object (cf Besley et al. [1993]). This in principle provides an efficiency rationale for auctories and would be consistent with seller’s expected revenue exceeding the retail price (because this outside option is not available to the budget–constrained agent). Secondly, the auctory may exploit behavioral biases of bidders by placing them in a very unfamiliar strategic environment. The benchmark Nash equilibrium calls for each agent to correctly estimate the probability of winning the auction, i.e. placing the lowest unique bid or not being overbid. In most cases, the equilibrium winning probabilities are very low. Prospect theory predicts overweighing of low probability events in decision making Kahneman and Tversky [1979]. This issue is exacerbated by the fact that a small absolute error can amount to a large absolute error around a low–probability event. Hence excess revenue for the seller may be large. There is a small but growing literature on auctories. Penny auctions are the most studied. Hinnosaar [2009] is the first study of the penny auction and presents the benchmark model. Augenblick [2009] develops a similar model and also analyses a dataset from Swoopo. He suggests that allowing for a sunk cost fallacy on the side of bidders can account for the high revenue also observed in his analysis. This paper seeks to contribute to the literature in two ways. First, we present evidence that expected revenue in auctories exceeds the value of the good by drawing on large datasets of both the penny and lowest unmatched bid auctions. We show that the

2

percentage excess revenue, defined as seller’s revenue less retail price, is independent of the retail value of the good but rises when the (equilibrium) probability of winning falls. In the penny auction, lower price increments are associated with greater excess revenue, and in the lowest unmatched bid auctions, bidding fees and excess revenue are negatively correlated. Second, we consider two candidate explanations of the stylised facts in a simple model.

2 Exploring Auctories 2.1 Penny Auction Data on almost 70000 auctions run on the UK site of penny auction platform Swoopo (http://www.swoopo.co.uk) are analysed in this section; summary statistics are presented in table 1. Goods sold on the site are largely new consumer good, and almost always branded goods; that is products, which would also be available at usual retail stores. The average retail value is GBP 159.89, and sells for just under GBP 27. Yet mean revenue for the seller is GBP 404.85, or more than 2.5 times the retail value of the good. The winning paid, after deducting the bidding fees paid, spent on average just shy of 40% of the retail value to secure the win of the auction. Bidding fees were constant at GBP 0.50 throughout the sample. Swoopo used several selling mechanisms during the sample period: • Standard penny auction: price increment 10p, winner pays final price (default) • Final price off auction: only bidding fees are paid, winner does not pay final price. The price counter is incremented by 1p per round, although this is payoff irrelevant. • Cent auction: price increment 1p only. Similarly 5p/20p auctions • Beginner Auction: only agents that have not won auctions before allowed to bid Moreover, auctions were manually categorised into groups where possible. Games consoles account for 30% of sales during the period, followed by consumer electronics – such as mp3 players and laptops – with 20%. Almost 15% of auctions involved vouchers of different kinds, such as gift cards for retailers, cash prizes or, indeed vouchers for further bids (such as “packs” of 150 bids). The key variable of interest is the profit rate, defined as (revenue/retail price)-1. Regression results for this variable are presented in table 1. Due to the very large dataset, all variables are statistically significant in each specification. Hence it is required to focus instead on the economic significance of the coefficient sizes. The leftmost model includes only the retail price as an explanatory variable; it is highly statistically significant but at a magnitude less than 1/1000 practically of small magnitude. The extremely low R-squared value confirms that the profit rate is not well explained by the retail price. This basic finding stays throughout all specifications; it suggests that the profit rate variable is appropriately scaled.

3

Model (2) adds auction characteristics. Ceteris paribus, a final price off auction raises the profit rate by a factor of 15 vis–a–vis the default. One–cent and two–cent auctions do similarly well, raising the profit rate by between 3.2–3.6. 5 cent auctions still yield 1.5 extra profit rate; only 20p auctions do somewhat worse than the 10p default price increment. Because a White test strongly rejects the null hypothesis, model (3) re-estimates the same equation using robust standard errors. Now all variables are significant at the 1% level. The final equation, model (4) adds product characteristics to the estimation. Notably, vouchers are by far the most profitable product category. Since they are also the most liquid products traded on the site, this suggests that a gambling motivation may be important in explaining bidding behaviour. Games consoles and then electronics are both slightly more profitable than uncategorised products. The equation was also estimated using robust standard errors, and again all variables are significant at the 1% level. Table 1: Penny Auction Summary Statistics Variable selling price retail price revenue winner claimed savings percent flag endprice flag onecent flag 2cents flag 5cents flag 20cents flag beginnerauction is voucher is electronics is console N

Mean 26.909 159.882 404.485 62.529 0.074 0.092 0.02 0.051 0.028 0.103 0.146 0.202 0.306

Std. Dev. 43.874 208.073 1257.325 30.159 0.261 0.288 0.141 0.22 0.164 0.304 0.353 0.402 0.461 68254

Figure 2.1 shows estimates hazard rates for 1640 auctions of 300 bid vouchers. These vouchers have a value of GBP 150, and were all carried out using the final price off rule. At a bidding fee of 50p, the equilibrium ending probability is 1/300 in each round – this rational bidders indifferent between bidding and staying out each period, see below –and is indicated in red. It is apparent that the estimated hazard rate is one order of magnitude below the predicted value of the baseline model.

2.2 Lowest Unmatched Bid Auction The data presented in this section were all collected from publicly available information on http://www.auctions4acause.com. A custom program was written that iterates through the past auctions available on the web site and extracts the information found

4

Figure 1: Regression Results (1) (2) (3) Retail Basic Robust profit rate profit rate profit rate retail price

-0.000691*** (0.000127)

-0.00266*** (0.000123) 15.53*** (0.0823) 3.200*** (0.0831) 3.619*** (0.156) 1.532*** (0.100) -0.290** (0.133) -0.171*** (0.0533) -1.261*** (0.0733)

-0.00266*** (0.000112) 15.53*** (0.262) 3.200*** (0.0929) 3.619*** (0.188) 1.532*** (0.0424) -0.290*** (0.0316) -0.171*** (0.0445) -1.261*** (0.0549)

1.828*** (0.0333)

0.729*** (0.0341)

0.729*** (0.0223)

flag endprice flag onecent flag 2cents flag 5cents flag 20cents flag click only flag beginnerauction is voucher is electronics is console Constant

Observations R-squared

68,254 68,254 68,254 0.000 0.354 0.354 Standard errors in parentheses *** p (V − f )/β (9) VS,i = −f 0 < i ≤ (V − f )/β   0 i=0 In period zero, the seller is indifferent over starting the auction. Due to the risk– neutrality, we can construct a simple indifference argument to see that expected

9

revenue must be equal to v conditional on sale taking place: Vsi |sale = 0 = ER − v

(10)

4. The result is immediate, because we are free to choose the probability of bidding in round zero. Hence ER = (1 − θ0 )v. This result is negative about the possibility for the seller to raise more revenue in the penny auction than using more standard mechanisms. Firstly, because the framework involves perfect information, the seller may as well make a take–it–or–leave–it offer with price V to any agent and collect the same revenue in equilibrium as in the best equilibrium of the penny auction. Moreover, the polar cases of war of attrition and ascending auction support the same outcomes in equilibrium as the penny auction: Claim 2 Equilibrium of WoA and Ascending Auction 1. The range of expected revenue that can be supported war of attrition equilibria is [0, V ] 2. Equilibrium expected revenue in the ascending auction is in the range [0, V ], but equal to V in the subgame perfect equilibrium. Proof

4 Adding Behavioural Bidders 4.1 Entertainment Value of Auctions Suppose payoffs in the standard game are slightly misspecified an omit a “joy of bidding” enjoyed by the participants in the penny auction. For example, ? suggests that bidders “pay a price for the thrill of the hunt” in the penny auction. Letting t be the value of this thrill, which is enjoyed irrespective of whether a bidder ends up winning the auction or not, gives rise to the period reward functions

u ˆ(B)

=

(θi+1 )(V − iβ) − f + t

(11)

u(O)

=

0

(12)

In the case of joy–for–bidding, it is clear that pay–per–bid auctions such as the penny auction yield superior revenue as compared to the ascending auction, because only in the former the seller can capture the surplus generated by the thrill from running the auction. By following the same steps as in the benchmark, we find that equilibrium ending probabilities are given by

10

θi =

  1

i > (V − f )/β 0 < i ≤ (V − f )/β i=0

f −t  v−iβ



0

Supported by bidding probabilities   0 f −t 1/(N −1) σi = 1 − [ v−iβ ]   1

i > (V − f )/β 0 < i ≤ (V − f )/β i=0

(13)

(14)

The per–period expected profit to the seller is: θi (iβ) + f = θi (v − v + iβ) + f = t +

v(f − t) v − iβ

which is positive as required. 4.1.1 Joy of Winning u ˆ(B)

=

θi+1 (V + j − iβ) − f

(15)

u(O)

=

0

(16)

Joy of winning simply to say that V from auction is higher than when buying from shop. Simple to show that expected revenue rises to V + j in all the cases we consider here - ascending auction, war of attrition, and penny.

4.2 Optimists and Pessimists Whether an agent is willing to participate in the auctory or not depends cruicially on her estimated probability of being overbid. The equilibrium of the baseline model relies on all agents estimating this probablity perfectly in each stage of the game. This ensures, in effect, that in each round of the penny auction, an actuarially fair lottery ticket is being sold. In this section, we relax this assumption and allow agents to make small but systematic mistakes in estimating their probability of winning in any given round. More precisely, we allow for Nb ≥ 2 behavioural bidders to be present in the auction who are optimists “optimists”– i.e. agents that slightly overestimate their probability of winning by an epsilon – and an equal number of “pessimists” that conversely underestimate their odds by the same amount. The resulting perceived probabilities of the different types are indicated in figure 4. Letting hats denote perceived values according to the agent’s type, this leads to perceived values of θi as follows:

11

Figure 4: Misperception: Over– and under–estimation by a fixed 

ˆ θi,O ˆ θi,R

= min(θi + b, 1)

ˆ θi,P

= max(0, θi − b)

= θi

Figure 4 illustrates this. The modified best response correspondence, with perceived values replacing actual ones, for type j is now  ˆ < f if θi+1  B V −iβ ˆ = f (17) brcj (θi+1 ) = {B, O} if θi+1 V −iβ   f ˆ O if θi+1 > V −iβ

We will now find the equilibrium when Nb ≥ 2. This assures there is always at least one non–highest bidder who is an optimist. To keep this agent indifferent between bidding and not bidding requires actual winning probability θi+1 =

f −b V − iβ

At this ending probability, rational and pessimistic agents strictly prefer to stay out of the auction. The per–period expected profit of the seller is now θi (iβ) + f = bV

12

which is again positive and larger than in the case of joy of bidding discussed previously for plausible parameter values. To see how a small amount of noise can make a this difference, consider the following example. Suppose there are 2 optimists and 2 pessimists (Nb = 4) and arbitrary large number Nr of rational bidders. The auction is of the “final price off” type, so β = 0, and suppose V = 500 and f = 1. In this case, the equilibrium with only rational 1 in each round. Hence in expectation the auction will run for players requires θ∗ = 500 500 periods, and expected revenue equals 500 ∗ f = 500, the value of the good. Now set b = 0.001, so that behavioural agents incorrectly estimate the probablity by 1/1000. It can be verified there exists an equilibrium in which the optimists take turns bidding, and each sets θ∗ = 0.001 in each period; rationals and pessimists abstrain from bidding. The expected duration of the auction now rises to 1000 periods, and expected revenue has doubled to the same amount. The impact is large for two reasons. First, because there are many rounds, errors that are neglible in individual rounds accumulate. Secondly, given the lower probability of ending in each round, the expected duration of the auction rises, redoubling the effect.

4.3 Revenue Comparisions Now we want to show that equilibrium expected revenue increases in the value of the good and falls with the price increment The comparsion is based on the fact the per–period payoff of the seller is strictly positive. Raising the value of the good lowers the probability of ending in the preferred equilibrium each period, leading to a first order stochastic dominant distribution over auction duration. Lowering β has the same effect. The profit of the seller is V0S

=

∞ X

[Πij=0 (1 − θj )][θj ∗ (v − iβ) + f ]

i=0 (v−f )/β

=

X

[Πij=0 (1 − θj )][bV ]

i=0