Deterrence in Sequential Contests: an Experimental Study∗

Deterrence in Sequential Contests: An Experimental Study∗

Arthur Nelson†

November 20, 2019

Abstract Many contests are sequential, with leaders making decisions ﬁrst, and followers observing those decisions and responding to them. The theory predicts that, unlike in standard Stackelberg duopoly settings, in two-player sequential contests the leader has no strategic advantage. How- ever, this is no longer the case for sequential contests with multiple leaders. Applications include political competition with two established parties and a possibility for a third party entry, or R&D competition with multiple incumbents and a new entrant. We conduct a lab experiment testing the equilibrium predictions for two- and three-player sequential contests, with the corresponding simultaneous contests as controls. Consistent with theory, we ﬁnd evidence of entry deterrence by leaders in the three-player sequential contest, but not in the two-player version.

Keywords: contest, sequential move, Stackelberg, deterrence, experiment JEL Classiﬁcation Codes: C92, D72

∗I thank Dmitry Ryvkin, Mark Isaac, Carl Kitchens, Jens Grosser, Luke Boosey, David Cooper, the participants of the 2017 Economic Science Association North America Meetings, the 2018 Southern Economic Association Meetings, the Experimental Economics Readings Group at FSU, and the NKU Center For Economic Education for their helpful comments. This research was done while the author was a Ph.D. student at Florida State University where he was supported in part by The Koch Fellowship and The Humane Studies Fellowship from the Institute for Humane Studies, whose support he gratefully acknowledges. †Department of Economics, Florida State University, [email protected].

1 1 Introduction

In contests, such as political campaigns, lobbying, or patent races, agents expend non-recoverable resources to secure a valuable prize. These contests often have a dynamic element to them, with agents making their decisions sequentially rather than simultaneously. For example, in the late 1970s Lockheed Martin and Northrup Grumman began working on stealth aircraft technology. After the technology was shown to work the Department of Defense solicited bids from the general defense contracting industry. The initial outlays to build facilities capable of producing these aircraft may have provided an advantage to the two companies. We observe candidates and political parties moving in a sequential fashion as well. In the 2016 Democratic primaries Hillary Clinton’s campaign announced well in advance of any voting that she had secured 20% of the ballots necessary to win the nomination.1 A very reasonable explanation for this announcement is that it was meant to discourage potential challengers in a potentially noisy selection process. Advantages in timing of decisions can also occur as a result of rules and regulations. The Libertarian Party had to undertake costly petition gathering and campaigning to be on the ballot in all states in 2016, announcing their success in doing so on September 8th, only two months before the general election. Decisions on where to invest effort for the Libertarian Party were undoubtedly influenced by the expenditures by Democrats and Republicans prior to this point. It is well-known from the industrial organization literature on oligopolistic competition among otherwise symmetric firms that, at least in theory, one should expect (1) a difference in behavior between settings where firms’ decisions are simultaneous and sequential; and (2) a strategic advantage of leader(s) in settings with sequential decisions (for a review, see, e.g., Ketchen Jr, Snow and Hoover, 2004). In political contests, this manifests in the form of higher expenditures and an incumbency advantage for established parties/candidates and can present a form of deterrence for non-establishment entrants. This may be a factor in the stability of the (effective) two-party system in the US.2 In this paper, we explore behavior in sequential two-player and three-player contests. We utilize a 2 × 2 between subject design to test theoretical point predictions and comparative statics. In the four treatments, we alter the number of players competing for the prize (n = 2 or n = 3), as well as the timing of decisions (simultaneous or sequential), where the sequential treatments involve one incumbent for n = 2 and two incumbents for n = 3. The three-player sequential treatment mimics a setting with two established contestants, such as two major political parties, and a possibility of a third-party participation. Contests are popular models that are widely used to describe a number of settings where resources are allocated competitively outside traditional markets. These include rent-seeking and

1See, for example, http://www.dailymail.co.uk/news/article-3214705/Hillary-one-fifth-delegates-ll- need-Democratic-nomination-locked-five-months-primary-vote-cast.html. 2Downs(1957) and Duverger(1959) predict a two-party equilibrium, but not necessarily with the same two parties, and the ﬁrst mover(s) advantage can help describe this phenomenon. See also Midlarsky(1984) and Meguid(2005) for a review of factors underlying the stability of the two-party system in the US.

2 other forms of political competition as well as R&D competition (Congleton, 1986). Particularly popular is the Tullock(1980) lottery contest model. Interestingly, when applied to a Stackelberg-like sequential move setting, this model predicts no difference in equilibrium expenditures between two- player sequential and simultaneous-move games (Linster, 1993). Thus, no deterrence is predicted in sequential Tullock contests with one leader and one follower. This counterintuitive result arises from the best response curves for both players intersecting at their maxima – a property that is unique to the two-player case. In contrast, we show that in equilibrium of the relevant three-player case with two leaders and one follower the leaders do have a strategic advantage. We experimentally test these predictions, focusing on evidence of entry deterrence. Consis- tent with theory, we find no evidence of entry deterrence or first-mover advantage in two-player sequential contests. In contrast, we do find some evidence of entry deterrence, with followers decreasing expenditures by approximately 10% relative to leaders in an otherwise symmetric setting, in three-player sequential contests. Entry deterrence is a difficult concept to disentangle from other strategic considerations in natural data. This is especially true in political competitions where unobservables, such as charisma or political savvy, are important determinants of candidates’ success. Laboratory experiments, with a much tighter control on incentives and other factors, can provide useful insights on the fundamental properties of behavior in such settings and represent an alternative research methodology. Our paper is one of the first to study experimentally behavior in sequential-move contests, and the first to explore behavior in sequential contests with multiple incumbents.3 The closest to this paper is the study by Fonseca(2009) who conducted an experiment with two-player sequential contests focusing primarily on the effect of asymmetric abilities, but also looked at the symmetric case as a control. Consistent with the prediction of Linster(1993), and also with our findings, (Fonseca, 2009) does not identify a difference in investment between leaders and followers in that treatment. The rest of the paper is organized as follows. Section2 is the theoretical model and resulting predictions, Section3 lays out the experimental design, Section4 reports the results, Section5 discusses several possible explanations for the observed deviations of subjects’ behavior from theory and implications for the model, and Section6 concludes.

2 Model and Predictions

2.1 Simultaneous Move Model

In the simultaneous move setting, n ≥ 2 identical risk-neutral agents compete for an indivisible prize V > 0. Each agent begins with an endowment ω > 0 and chooses a contest expenditure

3Distinct, but somewhat related to this is the substantial literature on dynamic contests. In these games, agents make decisions simultaneously and repeatedly, with or without feedback (e.g., Harris and Vickers, 1985; Konrad and Kovenock, 2009; Seel and Strack, 2013). Another type of dynamic contests are those consisting of stages, with losers eliminated and winners moving on to the next stage (e.g., Rosen, 1986; Gradstein, 1998; Moldovanu and Sela, 2006; Fu and Lu, 2012).

3 xi ∈ [0, ω]. The probability player i wins the contest is given by the lottery contest success function (CSF) of Tullock(1980):

 xi Pn Pn if j=1 xj > 0  j=1 xj pi(xi, x−i) =  1 Pn n if j=1 xj = 0 The winner of the contest receives the prize while all other players get zero. This gives the expected payoﬀ of player i

πi = ω − xi + pi(xi, x−i)V.

This game has a unique symmetric Nash equilibrium (e.g., P´erez-Castrilloand Verdier, 1992) ∗ with individual expenditure levels xi = xsim, i = 1 . . . , n, where

n − 1 x∗ (n) = V. (1) sim n2

Equation (1) implies the basic size eﬀect for simultaneous move contests: The equilibrium expenditure is decreasing in n.

2.2 Two-Player Stackelberg Model

In this setting, there are two agents who start with endowments ω > 0 and move sequentially.

Agent 1 (the leader) chooses her expenditure, xl ∈ [0, ω], first; agent 2 (the follower) observes xl and chooses her expenditure, xf ∈ [0, ω]. The payoffs are then determined in the same way as in the simultaneous move contest with n = 2. This game was first analyzed by Linster(1993). The equilibrium expenditures for the leader and the follower are the same, and equal to the symmetric equilibrium expenditure in the simultaneous move contest of two players: V x∗(2) = x∗ (2) = . (2) l f 4 This result is somewhat counterintuitive and follows from the properties of the best response function for Tullock contests; specifically, from the fact that at the symmetric equilibrium in the simultaneous move contest of two players the two best response functions intersect at their global maxima. This would not be the case, for example, in a standard Cournot doupoly model with linear demand, where the equilibrium output levels of Stackelberg leader and follower are distinct and different from the simultaneous move equilibrium.

2.3 Three-Player Stackelberg Model

Here, the setting is similar to the two-player Stackelberg model, except in the ﬁrst stage there are two leaders who make their expenditure decisions, xl1, xl2 ∈ [0, ω], simultaneously and independently. In the second stage, a single follower observes xl1 and xl2 and chooses her expenditure,

4 xf ∈ [0, ω]. The payoﬀs are then determined in the same way as in the simultaneous move game of three players.

Given the leaders’ decisions xl1, xl2, the expected payoﬀ of the follower is given by

V xf πf (xf ; xl1, xl2) = − xf , xf + xl1 + xl2 which produces the best response function for the follower in the second stage:

n 1 o br 2 xf = max (V (xl1 + xl2)) − (xl1 + xl2), 0 . (3)

Assuming the follower’s expenditure is positive, and substituting the best response into the payoﬀ function for leader 1 gives xl1 πl1 = 1 V − xl1. (4) (V (xl1 + xl2)) 2

This function is strictly concave in xl1; therefore, there is a unique interior maximizer. Assuming the two leaders behave symmetrically, the ﬁrst-order condition for payoﬀ (4), combined with (3), gives the equilibrium expenditures

9V 3V x∗ = x∗ ≡ x∗(3) = , x∗ (3) = . (5) l1 l2 l 32 f 16

As seen from (5), in the three-player Stackelberg model the equilibrium expenditures of the leaders and the follower are no longer equal, and the leaders have a strategic advantage. Thus, the equilibrium in the three-player Stackelberg model can be interpreted as one exhibiting entry deterrence by the two incumbents, which would also be expected in a standard Cournot oligopoly setting. Note that this is in contrast to the two-player case where the sequential nature of the game has no eﬀect on equilibrium expenditures, cf. Section 2.2. This matches the predictions in Hinnosaar(2018), which provides a solution method to sequential contests of any size.

3 Experimental Design and Hypotheses

3.1 Treatments

The experiment followed a 2 × 2 between-subject design, by varying the group size (n = 2, n = 3) and whether decisions were made simultaneously (SIM) or sequentially (SEQ). The resulting four treatments are denoted as SIM2, SIM3, SEQ2 and SEQ3.

3.2 Procedures

All experimental sessions were performed at Florida State University (FSU) during the Spring and Summer semesters of 2018. Subjects were FSU undergraduate students recruited randomly via email using ORSEE (Greiner, 2015). 270 subjects participated, 65.9% of them female. All

5 Treatment Sessions Subjects Matching Groups SIM2 3 54 9 SEQ2 4 54 9 SIM3 5 81 9 SEQ3 6 81 9 Total 18 270 36

Table 1: A summary of treatments. treatments were programmed and run using z-Tree (Fischbacher, 2007). Subjects received $21.15 on average, including a $7 show-up fee. Sessions lasted under 90 minutes. At the beginning, subjects received a package with all the instructions.4 The experiment con- sisted of three parts, and instructions for each part were read aloud at the start of that part. In Part 1, subjects’ risk attitudes were measured using the list elicitation method of Holt and Laury (2002). Subjects were presented with a list of ten choices between lotteries ($2.00, $1.60; p, 1 − p) (Option A) and ($3.85, $0.10; p, 1 − p) (Option B) where p varied between 0.1 and 1 in increments of 0.1. For each p, subjects indicated their preference for Option A or Option B. One of the ten choices was selected randomly and played for actual earnings, with the results and payoffs withheld until the end of the session. In Part 2 – the main part of the experiment – subjects competed in a series of contests. Af- ter receiving the instructions, subjects went through a practice round where they could test the mechanics of the game without interacting with each other. Subjects were provided with examples and then allowed to make investment decisions according to the treatment they were in, for themselves and hypothetical others, and observed the resulting probabilities of winning. To ease the computational burden on subjects, a calculator was built into the program to allow subjects to calculate probabilities using the lottery CSF. This calculator was available to subjects at all times during Part 2. After practice, subjects were randomly divided into non-interacting matching groups of 6 (for n = 2 treatments) or 9 (for n = 3 treatments). In the SEQn treatments, subjects were randomly assigned to the roles of leaders and followers and remained in those roles throughout the session. There were 25 rounds with random re-matching in groups of n within the matching groups. In each round, subjects received an endowment of 240 experimental points and chose their investments.5 The prize was 240 points. In SIMn treatments, all subjects in the group chose investments simultaneously, while in SEQn treatments the leader(s) chose their investment first, which were observed by the follower, who then chose his or her investment. At the end of the round, subjects were informed if they won the contest and observed the resulting payoff for the round. One of the rounds was chosen randomly for actual earnings, with points converted to US dollars at the

4Instructions for treatment SEQ3 are provided in AppendixA. Instructions for the other treatments are similar, and instructions for the Holt and Laury(2002) task are standard. Both are available upon request. 5Investments were restricted to integer numbers in [0, 240]. Under this restriction, and given the parameters of the game, the equilibria identiﬁed in Section2 are still valid.

6 exchange rate of $1=20 points.6 The experiment concluded with Part 3, where subjects completed a short demographic questionnaire, reporting what math and statistics courses they had taken and how competitive they believed they were compared to an average person.

3.3 Hypotheses

The focus of this study is on the strategic advantage of the first movers (leaders) in the SEQ2 and SEQ3 treatments, which can be interpreted as a form of deterrence by incumbents. The simultaneous-move treatments serve as baselines. Below, we formulate three hypotheses that follow from the equilibrium predictions in Section2. The first hypothesis describes the basic group size effect comparing investments in SIM2 and SIM3. From (1), the individual investment is decreasing in group size. This effect is common in contests and other competitive environments, such as Cournot oligopoly. The intuition is that when the number of competitors increases, each individual’s probability of winning, and hence the marginal benefit of investment, decreases.

H1 (Group size eﬀect): The individual expenditures will be lower in the Simultaneous treatment with n = 3 than in the Simultaneous treatment with n = 2.

The evidence regarding the effect of group size on expenditures from previous contest experiments is inconclusive. Baik et al.(2015), comparing contests with group sizes 2 and 3, as well as Lim, Matros and Turocy(2014), comparing group sizes 2, 4 and 9, find no effects of group size on average expenditures, although the latter study finds an increase in variance. At the same time, Anderson and Stafford(2003) and Morgan, Orzen and Sefton(2012) find, consistent with the equilibrium predictions, a reduction in average expenditures with group size. It should to be noted, however, that these two studies allow for endogenous entry; therefore, if restricting attention only to experiments with fixed group sizes, there is no strong evidence to support the predicted group size effect in prior experiments. The next hypothesis describes a comparison between SIM2 and SEQ2, where the equilibrium expenditures coincide.

H2: Individual expenditures of leaders and followers in SEQ2 will be the same, and equal to equilibrium expenditures in SIM2.

The only experimental study we are aware of where expenditures in simultaneous-move and sequential-move two-player contests are compared is by Fonseca(2009). The focus of that paper is on the effects of heterogeneous abilities, but it appears from the homogeneous baseline data that average payoffs of leaders and followers were not statistically significantly different from payoffs in

6At the beginning of the experiment subjects were informed that the round they would be paid for had been previously drawn using a random number generator. This number was written down and put in an envelope at the front of the room for transparency. The envelope was opened and the paying round was revealed on the subjects’ screens at the end of the experiment for comparison.

7 the simultaneous-move setting. This may indicate that leaders’ and followers’ average expenditures were also similar between themselves and to the simultaneous-move expenditures, consistent with theoretical predictions. The ﬁnal hypothesis stems from the comparison of leaders’ and followers’ equilibrium expenditures in SEQ3, and a comparison of both of those to expenditures in SIM3.

H3: In SEQ3, the follower will have a lower expenditure than the leaders, with the SIM3 expenditures in between the two.

Hypothesis H3 characterizes deterrence behavior by the leaders in SEQ3. H2 and H3 combined predict a qualitative diﬀerence between sequential-move settings with one and two leaders. The presence of two leaders (incumbents) restores the strategic advantage of ﬁrst movers that is absent in SEQ2 due to the special structure of best responses in the Tullock contest.

4 Results

Table2 presents average expenditures and payoffs by treatment, subdivided by role where appropriate. The table also shows predicted (equilibrium) expenditures and payoffs and p-values for comparisons of observed and predicted levels for both variables. Consistent with the results of most previous studies (see, e.g., surveys by Sheremeta, 2013; Dechenaux, Kovenock and Sheremeta, 2015), we observe overbidding in every treatment and role. The only two instances where overbidding is not statistically significant are SIM2 and leaders in SEQ3. Overbidding has consequences for subjects’ payoffs. Specifically, with the exception of SIM2 and followers in SEQ3, payoffs are significantly lower than predicted. While in SIM2 both expenditures and payoffs are in line with the equilibrium predictions, in SEQ3 leaders’ payoffs are below equilibrium even though their expenditures are not different from equilibrium, due to the significant overbidding by followers. To verify that the random assignment of subjects to treatments was balanced, we conducted Pearson’s χ2 nonparametric test comparing the distributions of demographic characteristics across treatments. We used subjects’ records in the ORSEE recruitment system to identify their Gender – a binary variable equal 1 for males and 0 for females – and Econ – an indicator of whether the subject’s reported major is listed as Economics.7 Risk T olerance, collected in the first part of the experiment, is the number of risky choices made in the Holt and Laury(2002) task. Competitiveness, collected in a post-experimental questionnaire, is a self-reported measure of how competitive the subject believes they are relative to average using a 1-5 Likert scale. The results are reported in Table3 and show that the assignment to treatments was balanced.

7The major of some subjects was listed as “Undergraduate” without a speciﬁed ﬁeld of study; for these subjects, the value of Econ is missing.

8 Treatment Role Round Payoff p-value Expenditures p-value Observed Predicted for diff. Observed Predicted for diff. SIM2 311.2 300 .24 70.49 60 .26 (8.83) (8.56)

Leader 248.88 300 .0003 100.98 60 .001 SEQ2 (8.52) (8.2) Follower 270.88 300 .02 99.26 60 .001 (10.15) (7.74)

SIM3 233.67 266.67 .001 84.85 53.3 .001 (6.43) (6.29)

Leader 244.80 262.5 .02 75.73 67.5 .29 SEQ3 (6.35) (7.24) Follower 250.63 255 .46 68.30 45 .01 (5.623) (7.39)

Table 2: Summary statistics, standard errors clustered by matching group in parentheses. The p- values are for two-sided t-tests comparing the observed means to predictions, with errors clustered by matching group. 100 80 60 40 Average Expenditure 20 0 0 5 10 15 20 25 Period

SIM2 SIM3

Figure 1: Average expenditures by period for SIMn treatments. The horizontal lines show the corresponding Nash equilibrium levels.

9 Variable χ2 p Gender .73 .87 STEM 1.81 .61 Risk Tolerance 19.06 .87 Competitiveness 13.09 .36

Table 3: Pearson’s χ2 test comparing the frequencies of assignment to treatments for each variable.

4.1 Comparisons Across Treatments

We start by analyzing the basic group size eﬀect comparing the simultaneous-move treatments SIM2 and SIM3. Figure1 shows the average expenditures by round for the SIM n treatments. The horizontal lines show the corresponding Nash equilibrium expenditures. We can see that expenditures in both treatments start out around the same level, but in later periods there is a visible downward trend towards equilibrium in SIM2, whereas expenditures in SIM3 stabilize at a higher level. This is supported by the nonparametric Mann-Whitney (rank sum) test comparing distributions of average expenditures in SIM2 and SIM3, treating each matching group as one independent observation (p = .0243).

Result 1 (Group size eﬀect in SIMn): Average individual expenditures are lower in SIM2 than in SIM3.

Result 1 is contrary to the theory and most experimental findings from previous literature (cf. Section 3.3). Especially remarkable is the absence of overbidding in SIM2 in later periods. One possible explanation for behavior in SIM2 is a wealth effect stemming from the relatively high budget of 240 points, as compared to the equilibrium prediction of 60. Baik, Chowdhury and Ramalingam(2015) show that increasing the contest budget produces an inverted-U change in contest expenditures, with a relatively small and relatively large budgets both leading to lower expenditures than an intermediate budget. That said, the same effect should be present in SIM3 where the equilibrium is even lower, but it is not, leading to the reversal of the predicted group size effect. One possible explanation can be differential overbidding due to differences in expectations about winning, termed “constant winning aspirations” (Boosey, Brookins and Ryvkin, 2017). The conjecture is that subjects value winning more when it is harder to win, leading to a larger joy of winning in SIM3 than in SIM2. Next, we turn to comparing expenditures in SIM2 and SEQ2. According to theory, there should be no difference between the two, as well as no difference in behavior of leaders and followers in SEQ2. The left panel in Figure2 shows the empirical CDFs of individual expenditures in SIM2 and separately for leaders and followers in SEQ2. As seen from the figure, expenditures in SEQ2 appear to first-order stochastically dominate those in SIM2, without obvious differences between the distributions for leaders and followers in the former treatment. This observation is confirmed by pairwise rank-sum tests, see the n = 2 row in Table4.

10 eut3 Result marginally ( a SEQ3 but in SIM3, followers in 4). subjects and in and SIM3 SEQ3 between expenditures between in differences leaders difference individual the significant between significant between and of statistically differences SEQ3 no CDFs clear in reveal followers no empirical and are tests leaders the There rank-sum shows SEQ3. Pairwise in 2 followers distributions. and Figure three between leaders in in for be panel separately should and SIM3 right SIM3 in The expenditures and SEQ3, two. in the followers than more spend should leaders behavior. deterrence of form any and observe not by leaders’ do driven We whereas be level (2009). experiment, to same seems the the again of difference about subjects. half SIM2 the at of Thus, first out behavior the stable. start anomalous the remain curves in also SEQ2 (and three falling in figure the are expenditures the ), 1 followers’ SIM2 from Figure in seen of As expenditures discussion but time. the over in change followers) earlier and noted leaders for (separately SEQ2 2 Result b ujcsi I3hv ihrepniue hnfloesi SEQ3. in followers than expenditures higher have SIM3 in Subjects (b) with treatments the of analysis the to Fonseca of turn experiment the we with Finally, also and theory, with consistent is 2 Result of (b) part However, hypothesis against different. goes not (a) are SEQ2 Part in follower and leaders of expenditures Average (b) a nSQ h olwradlaesaesaitclyindistinguishable. statistically are leaders and follower the SEQ3 In (a) : SEQ2. in than lower are SIM2 in expenditures individual Average (a) : iue2 miia Dso ujcs xedtrsb lyrtype. player by expenditures subjects’ of CDFs Empirical 2: Figure Empirical CDF 0 .5 1 0 100 n=2 H2 200 Simultaneous Leader soshwaeaeepniue nSM and SIM2 in expenditures average how shows 3 Figure . Expenditures 11 300 0 p 100 Follower n 0 = .Acrigt hoy( theory to According 3. = n=3 . 7 e the see 07, 200 n 300 o nTable in row 3 = H3 ), iue4 vrg xedtrsb eidfor levels. period equilibrium by Nash expenditures corresponding Average 4: Figure for level. period equilibrium by Nash expenditures corresponding Average 3: Figure

Average Expenditures Average Expenditures 0 25 50 75 100 0 25 50 75 100 125 0 0 5 5 Simultaneous Leader Simultaneous Leader 10 10 n n 12 Period Period ramns h oiotllnsso the show lines horizontal The treatments. 3 = the shows line horizontal The treatments. 2 = 15 15 Follower Follower 20 20 25 25 Leader vs. Follower Leader vs. Simultaneous Follower vs. Simultaneous z p-value z p-value z p-value n = 2 -.221 .8253 2.075 .0380 2.252 .0243 n = 3 -.927 .3538 -.927 .3538 -1.810 .0703

Table 4: Results of rank-sum tests for comparisons between distributions of expenditures in SIMn and SEQn (separately for leaders and followers). Each matching group is treated as one independent observation, with 9 independent observations per role.

Part (a) of Result 3 does not confirm the theoretical prediction. Leaders’ expenditures are slightly higher than followers’, but the effect is smaller than predicted, and too noisy for statistical signif- icance. As seen from Table2, the predicted difference between leaders and followers in SEQ3 is 50% (67.5 vs. 45). The observed difference is only 11% (75.73 vs. 68.30). Thus, we do not observe the predicted deterrence behavior in SEQ3. Part (b) of Result 3 is consistent with theory, but this is mainly due to excessive expenditure in SIM3. Figure4 shows average expenditures in SIM3 and SEQ3 (separately for leaders and followers) by round, along with the equilibrium predictions. Interestingly, while subjects in SIM3 as well as followers in SEQ3 remain above equilibrium, the investment of leaders in SEQ3 stabilizes very close to its equilibrium level. That explains the absence of deterrence. Indeed, if all subjects were overinvesting in the same way, the comparative statics in H3 would have held, at least directionally. Leaders are the only subjects whose investment is directly observed (by followers). In section 4.2, we explore in detail how followers react to this information within a round, as well as how all subjects respond to the information they receive between rounds.

4.2 Dynamics

Table5 reports the results of dynamic regressions for individual expenditures. Columns (1)-(3) combine treatments SIM2 and SEQ2, while columns (4)-(6) combine treatments SIM3 and SEQ3. F ollower and Leader are indicator variables for the two roles in SEQn. The SIMn treatments serve as baselines. Columns (1) and (4) report regressions with basic treatment eﬀects, reproducing Results 2 and 3. As expected, there is a signiﬁcant negative time trend in both cases. Columns (2) and (5) additionally control for the subject’s own expenditure in the previous round

(Expendituret−1) and an indicator of whether the subject won the contest in the previous round

(W int−1). The effect of the lagged expenditure is positive and statistically significant, indicating strong persistence (or “fixed effect”) in individual investment decisions. Winning in the previous round has a negative effect on expenditure for n = 2, which is typically interpreted as evidence of reinforcement-type learning. Adding interactions of W int−1 with roles reveals that the effect of W int−1 is only significant in SIM2. That is, subjects in SEQ2 do not significantly adjust their expenditures in reaction to winning in the previous round. This difference in dynamics explains why average expenditures end up being lower in SIM2 as compared to SEQ2. Winning last period is weakly significant in the SIM3 treatment. Followers in SEQ3, however,

13 n = 2 n = 3 (1) (2) (3) (4) (5) (6) Follower 28.77** 6.621 1.991 -16.55* 4.273 -0.357 (11.20) (7.289) (8.596) (9.418) (3.313) (4.319)

Leader 30.49** 5.516 0.479 -9.119 -2.288 -5.061 (11.50) (5.249) (5.761) (9.312) (2.930) (3.582)

Period -1.300*** -0.576*** -0.553*** -0.712*** -0.210** -0.0989 (0.198) (0.103) (0.139) (0.180) (0.0928) (0.116)

Wint−1 -13.21*** -18.04*** 5.838* 3.722 (3.641) (5.538) (2.994) (3.537)

Follower×Wint−1 3.761 13.37 -28.24** -29.05*** (10.16) (11.95) (10.45) (9.549)

Leader×Wint−1 8.571* 10.29 -0.736 3.641 (4.921) (6.556) (4.187) (5.866)

Expendituret−1 0.667*** 0.644*** 0.709*** 0.683*** (0.0453) (0.0519) (0.0332) (0.0414)

Male -3.031 -3.377 (3.253) (4.165)

Econ -4.442 2.672 (3.425) (3.101)

Competitive -1.997 2.327* (2.112) (1.138)

RA 1.051 2.873*** (1.227) (0.922)

Constant 87.39*** 38.38*** 48.19*** 94.11*** 25.31*** 9.995 (8.953) (6.586) (13.73) (7.155) (3.818) (6.465) N 2700 2592 2112 4050 3888 2880

Table 5: OLS regressions of expenditure on various controls, with standard errors clustered at the matching group level. Signiﬁcance levels: * p < 0.10, ** p < 0.05, *** p < 0.01.

14 strongly react to winning in the previous period, with a nearly 30 point drop in expenditures following a victory. This would help explain the drops in expenditures for followers seen in Figure 4. One possible explanation for the difference between n = 2 and n = 3 players reactions to winning is the information received at the end of the period. The subject is always told whether they win or not and their payoff. In SIM2 a victory gives weak information that the subject is likely to be the higher bidder, with the same logic holding for losers and low bidding. Independently adjusting to the mean to try and maximize payoffs would generate the SIM2 result. In the n = 3 treatments the logic is much more complicated, winning may still indicate overbidding, but losing no longer has a meaningful signal, as you could be the median bidder, or the low bidder. This is a simplistic view by subjects, as the highest bidder does not always win, but it is still a type of analysis one might expect in a contest. Regressions in columns (3) and (6) additionally control for subjects’ individual characteristics (Male and Econ are binary, Competitive is on the 1-5 Likert scale, and Risk T olerance is between 0 and 10). These additional controls have little effect on coefficients beyond what the dynamic terms provide. Reassuringly, subjects who view themselves as more competitive and/or have higher tolerance for risk bid more on the contests on average, as one would expect. The coefficient on Period ranges from negative and insignificant to a statistically significant half point plus decline in expenditure each period. Table9 in AppendixB reports the results from the last 15 periods, with Period insignificant in each regression, it seems that the majority of this decline is early on, consistent with learning. Turning our attention to how subjects’ strategies develop over time, we see evidence of some learning. Figures3 and4 combined show average expenditures per period for every player type and number of player combination. The general trend seems to be negative, given that subjects overbid, and even more so in early periods over the course of the 25 periods, they develop the knowledge that they can reduce expenditures and get higher payoffs. This is confirmed in Table5, which reports the results of regressions run on individual expenditures. We see evidence of persistence in expenditures over time from Table5, consistently players spend about 70 points in a period for every 100 points spent in the previous period. These coefficients are highly significant and robust to additional controls. Winning last period drops expenditures significantly in the n = 2 treatments, but has no effect in the n = 3 treatments.

5 Discussion

As evidenced in the previous section, the observed behavior deviates substantially from equilibrium predictions. Most notably, we do not observe the predicted differences between leaders’ and followers’ expenditures in SEQ3 where we expected deterrence. At the same time, across the n = 2 treatments we observe differences where theory predicts we should not. We know from previous studies that our SIM2 behavior is anomalous, in that there is no significant overbidding (Sheremeta,

15 n = 2 n = 3 (1) (2) (3) (4) (5) (6) Total Leaders’ Expenditures -0.535** -0.402 -0.454*** -0.400* (0.211) (0.286) (0.116) (0.174)

Square Root of Total Leaders’ Expenditures 10.43** 8.116 8.818*** 8.452** (3.963) (5.487) (1.956) (2.779)

Max of Leaders’ Expenditures -0.108** 0.0226 (0.0458) (0.0795)

Min of Leaders’ Expenditures -0.0136 -0.0682 (0.111) (0.0941)

Period -0.568 -0.602 -0.413 0.147 -0.430 0.0399 (0.389) (0.428) (0.341) (0.441) (0.349) (0.450)

Expenditurest−1 0.435*** 0.446*** 0.368*** 0.189** 0.370*** 0.203** (0.0756) (0.0845) (0.101) (0.0748) (0.101) (0.0765)

Wint−1 3.145 4.484 -1.843 -2.692 -1.500 -3.496 (10.06) (10.47) (7.039) (8.076) (6.694) (6.761)

Male 4.527 -14.37** -15.70** (15.69) (6.172) (6.178)

Competitive 0.924 -2.632 -1.253 (4.618) (5.127) (4.593)

Risk Tolerance 0.712 6.263 6.790 (3.385) (5.303) (5.004)

Econ -25.74 -0.709 56.31*** (17.00) (5.067) (11.27)

Econ × Max -0.525*** (0.0751)

Constant 20.81 23.37 16.44 0.625 61.84*** 30.76 (25.82) (46.04) (14.75) (17.87) (10.36) (19.86) N 648 576 648 456 648 456 Standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01

Table 6: OLS regressions for followers’ expenditures in SEQn, with standard errors clustered at the matching group level. Signiﬁcance levels: * p < 0.10, ** p < 0.05, *** p < 0.01.

16 2013). However, SIM2 only serves as a baseline, and we chose not to explore these anomalies in more detail. The main focus of this paper is on deterrence in the SEQn treatments, which we address below.

5.1 Stackelberg Follower Behavior

Table6 shows the results of regressions of followers’ expenditures in the SEQ n treatments. Columns (1) and (2) refer to SEQ2, whereas columns (3)-(6) refer to SEQ3. In each case, a specification with different feedback variables is accompanied by the same regression with additional controls for individual characteristics. Columns (1) and (2) for SEQ2, as well as columns (3) and (4) for SEQ3, show the dependence of followers’ expenditure on the aggregate behavior of leaders in the same period. Variable Total Leaders’ Expenditures is defined as the leader’s expenditure in SEQ2, and the sum of both leaders’ expenditures in SEQ3. From equation3 it can be calculated the best p p response for the followers is given by: 240 ∗ (x1 + x2) − (x1 + x2) ∼ 15.5 (x1 + x2) − (x1 + x2) the coefficient on Square Root of Total Leaders’ Expenditures is not significantly different from the predicted value of 15.5 for the n = 2 treatments. This would indicate that in simpler contexts, i.e. only one other competitor the revelation of information in a sequential contest reinforces the tendency towards equilibrium. In n = 3 treatments the additional player’s information is not fully utilized, so the followers react less intensely to leaders’ expenditures than equilibrium predicts. In SEQ3, it may also be of interest to explore whether followers react to the behavior of the two leaders separately. Columns (5) and (6) include variables Max of Leaders’ Expenditures and Min of Leaders’ Expenditures defined, respectively, as the higher and the lower of the two leaders’ investments. Column (5) shows that followers react negatively to the larger expenditure. This suggests that followers perceive competition to be driven by the stronger of the two opponents. This is at odds with their best response that is determined by the sum of leaders’ expenditures. It turns out that the differential reaction to the two leaders is driven primarily by subjects who are Economics majors. Column (6), in addition to the usual individual characteristics, also includes the interaction Econ×Max between the Econ dummy and the maximum leader’s expenditure. The coefficient on Econ is 56.31 and highly significant, whereas the coefficient on the interaction is −0.525, also highly significant. At the same time, the coefficient on Max of Leaders’ Expenditures is no longer significant. This shows that when leaders are weak, Econ followers react with large investments, but as the maximum of the leaders’ investment increases, Econ followers reduce their investment. In contrast, non-Econ followers are not affected by Max of Leaders’ Expenditures at all. These responses of Econ followers are consistent with best response and deterrence behavior at least to some extent, albeit they react only to the maximum. Next, we turn to the analysis of deterrence by comparing the behavior of leaders and followers. Columns (1)-(4) in Table7 show that there are no differences in expenditure between leaders and followers in SEQ2 for non-Econ students; however, Econ leaders invest more than non-Econ leaders, whereas Econ followers invest less than Econ leaders8 and less than non-Econ leaders. These results 8p = 0.09 for the Wald test for the sum of coefficients on Econ and F ollower × Econ in column (2).

17 n = 2 n = 3 (1) (2) (3) (4) (5) (6) (7) (8) Follower 4.012 9.345 1.368 4.384 -18.27 -17.61 -6.894* -6.274* (12.16) (13.05) (6.159) (6.774) (9.897) (9.592) (3.421) (3.059)

Econ -5.141 50.59*** -2.992 27.63** 1.471 4.276 -0.151 2.448 (7.986) (11.99) (4.027) (8.489) (14.83) (22.80) (6.132) (9.114)

Follower×Econ -84.52*** -46.53*** -7.153 -6.620 (17.28) (9.694) (24.23) (8.795)

Wint−1 -0.0634 -0.645 3.138 3.060 (5.828) (6.229) (4.006) (3.974)

Expendituret−1 0.533*** 0.523*** 0.622*** 0.622*** (0.0720) (0.0710) (0.0526) (0.0528)

Period -0.757*** -0.766*** -0.209 -0.209 (0.198) (0.201) (0.141) (0.141)

Constant 98.91*** 95.97*** 56.51*** 56.25*** 77.53*** 77.32*** 30.76*** 30.58*** (12.22) (11.99) (7.543) (7.558) (6.092) (5.987) (3.656) (3.473) N 1075 1075 1032 1032 1475 1475 1416 1416

Table 7: Bidding behavior in SEQn treatments. Standard errors clustered at matching group level in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01 are robust to the inclusion of dynamic controls for own expenditure and winning in the previous round. Columns (5)-(8) in Table7 show the results of similar regressions for SEQ3. Here, there is some evidence of deterrence in dynamic regressions, and there are no differences between Econ and non-Econ subjects. Looking back to Table2, we see that the difference in mean expenditures between leaders and followers in SEQ3 is about 7 experimental points, about 10% of the leaders’ total expenditure. While this difference is not statistically significant, it is economically significant. The results in Table7 provide further evidence of deterrence in SEQ3, contrasted by only limited evidence of such deterrence in SEQ2 (observed only among Econ students). While not as strong as we would like to have found, in totality there does seem to be evidence of followers being deterred in the SEQ3 treatment, supporting the theoretical model.

5.2 Joy of Winning Models

Joy of winning (JoW), or non-monetary utility derived from winning a contest, is commonly used as an explanation for overbidding in contests (Delgado et al., 2008; Goeree, Holt and Palfrey, 2002; Sheremeta, 2010, 2015). We utilize and compare two models of the joy of winning. The ﬁrst, which we call the traditional model, uses an additive constant utility of winning w that is the same w(n−1) across treatments. The level of overbidding in this model is δxsim(n) = n2 for SIMn and SEQ2

18 treatments. For SEQ3, overbidding by leaders and followers, respectively, is (cf. (5))

9w 3w δx (3) = , δx (3) = . l 32 f 16

The second model relies on the assumption of “constant winning aspirations” (CWA) (Boosey, Brookins and Ryvkin, 2017). The idea behind CWA is that people enjoy winning more the more diﬃcult or unlikely it is. A simple speciﬁcation proposed by Boosey, Brookins and Ryvkin(2017) has the joy of winning decreasing linearly in the probability of winning; hence the expected non- monetary value of winning, W , is constant across treatments.

Let Pa denote the probability of winning in equilibrium, and let wa denote the resulting joy of winning for each player type a = s, l, f ([s]imultaneous, [l]eader, [f]ollower). Then the CWA assumption implies that W = Pawa for all a, and we can calculate overbidding due to joy of 1 CWA winning using equations (1) and (5). For SIMn, Ps = n and overbidding is simply δxsim (n) = ws(n−1) W (n−1) W (n−1) 2 = 2 = . For SEQ2, it is the same as for SIM2 for both leaders and followers. n Psn n Finally, for SEQ3 we have from Eqn. (5),

CWA CWA 9W 3W 3W δxl (3) = δf (3) = = = . 32Pl 16Pf 4

In both models, predicted overbidding levels are proportional to a parameter. For example, in the traditional model δx = kw, where k depends on the treatment and type. We estimate w by running a regression of observed overbidding on the corresponding coeﬃcients k combining data from all treatments. Similarly, W can be estimated for the CWA model. For this exercise, we used data from round 15 onward, to focus on decisions of experienced subjects. The resulting estimates are w = 65.69 (15.275) for the traditional model, and W = 25.69 (5.454) for the CWA model, with standard errors in parentheses. Detailed results are presented in Table8. Column (1) provides the probability of winning if each subject in the group plays their NE. Column (2) calculates the JoW adjusted for the probability of winning, wa. Column (3) reports the average bid for each role in period 15 onwards. Columns (4) and (5) are the calculated equilibrium bids utilizing the appropriate JoW parameter. Column (6) is the Nash equilibrium prediction, previously reported, repeated here for ease of comparison. The traditional and CWA models predict expenditures on the contest equally well on average, whether using MSE from regressions of expenditures on predicted expenditures (71.05 and 70.98), sum of deviations of predictions from observation (-16.91 and -16.47), R2 (0.0481 and 0.0501), or any other measure we considered. Notably, the CWA model of joy of winning ﬁts the observed behavior very well for SEQ3 followers. The utility of winning against two opponents who have a timing advantage is large and would partially explain the lack of separation between SEQ3 leaders and followers.

19 Treatment Role NE CWA Parameter Observed Predicted Expenditures Prob(win) by Role (wa) Expenditures CWA Traditional NE (1) (2) (3) (4) (5) (6) SIM2 0.5 51.38 61.51 72.85 76.42 60

Leader 0.5 51.38 89.93 72.85 76.42 60 SEQ2 Follower 0.5 51.38 91.65 72.85 76.42 60

SIM3 0.333 77.07 79.3 70.46 67.93 53.33

Leader 0.375 68.51 70.86 86.77 85.98 67.5 SEQ3 Follower 0.25 102.76 63.71 64.27 57.32 45

Table 8: Winning probabilities, observed and predicted average expenditures and estimated wa

5.3 Cognitive Complexity

The SEQ3 leaders have arguably the most cognitively taxing role in this experimental setup. These subjects have to consider what another subject is doing simultaneously, as well as how the follower will react to their decision. It would be logical for subjects with this level of complexity to attempt to simplify the game somehow. One possibility is for them to ignore the follower completely and instead treat the contest as a two-player simultaneous move game. Figure5 shows the empirical CDFs of expenditures of leaders in 3SEQ and players in SIM2 overlaid. As seen from the figure, and confirmed by a rank-sum test (z = 0.221, p = 0.8253), one would fail to reject the null that the distributions are the same. While there is no way to be certain that the SEQ3 leaders are following the strategy of SIM2 players in this design, it is a feasible explanation given the likely cognitive load inherent in their task. Deck and Jahedi(2015) provide a literature review of experiments on cognitive load one result being that as cognitive load increases subjects tend to anchor their behavior and rely on simple heuristics more for decision-making. As seen from Table5, leaders had no more responsiveness to previous round expenditures or winning, relative to simultaneous players, whereas followers were highly reactive to both variables. This would indicate some level of anchoring, by leaders who have a more complex environment to consider, but in theory, can affect followers bids by changing their own expenditures.

6 Conclusion

Many contests in the ﬁeld are sequential. Examples include political campaigns where contenders announce their participation at diﬀerent times, and where third-party candidates may enter races with established incumbents who always participate by default. Similar sequential investment decisions are frequent in R&D competition.

20 eiini ots.Ti ol matfim rigt e euaoyapoa oetro expand or enter to approval regulatory get to contracts-favors-ula-spacex/ trying firms impact could This contest. to a relative in disadvantage petition by competitive a that at claimed are has they contracts funding Force of Air companies. round aforementioned other first the has the Origin in which Blue included SpaceX, and being Grumman, dollars. not Northrup billion Alliance, 2 Launch over United totaling to Agreements Service Launch probability for higher a have In entrant. and the more, entry. than bid third-party advantage, equilibrium for strategic in applications a winning have of to incumbents relevant two are three-player the in which case, present this leaders, longer point two no maximum is with equivalence the contests this at However, sequential exactly lottery functions. falls two-player response game in move best responses simultaneous individual best the of of in structure equilibrium special sequential the the the to where between due contests behavior is This bidding equilibrium games. and simultaneous advantage, (subgame-perfect) no and in has players differences leader three the no or contests, are follower) sequential two-player there one For ques- and this follower). leader and one to (one advantage and answer players leaders strategic two the (two involving a in contests have differences (1980 ) leaders Tullock predicts the for theory extent tion The what strategies. to bidding is pre-emptive contests employ sequential about ask to tion simulta- in players to game sequential player 3 in leaders by game expenditures neous of Comparison 5: Figure 9 See hne oteifrainaalbet atcpn a togyiflec h ee fcom- of level the influence strongly can participant a to available information the to Changes Force Air US the by issued contracts 2018 October the is advantage such of example One ques- natural a organization, industrial in competition Stackelberg on literature the to Parallel https://arstechnica.com/science/2019/03/looming-air-force-decision-on-mid-2020s-launch-

CDF of Expenditures 0 .2 .4 .6 .8 1 0 . 9 50 3 PlayerLeader 100 Expenditures 21 150 2 PlayerSimultaneous 200 250 in a market, states receiving campaign dollars during primaries, sports teams getting a bye week, etc. From the above example, it would not be surprising if SpaceX changed how they compete for contracts going forward, now that they have information on their competitors’ technological capabilities. SpaceX has already asked California’s Senators to intercede in this case on their behalf. In this paper, we study experimentally behavior in sequential two-player and three-player contests. We provide the first experimental test of the theory for sequential three-player contests with two incumbents and find evidence, albeit weak, for entry deterrence. As controls, we also provide the first direct comparisons between simultaneous-move and sequential-move contests with two and three players. Consistent with theory, we find no evidence of entry deterrence in the two-player case. The results suggest that in winner-take-all contests, such as U.S. elections, at least two participants should always move first. This is because any attempt to preempt the second party results in a lower expected payoff for the leader in the two-player case. We also find that third parties are suppressed in the competition, but are not completely eliminated. The reasons for third party persistence in elections, beyond the probabilistic nature of the competition, are beyond the scope of this paper. The analysis presented in this paper is the most obvious first step to identifying the role of incumbency in sequential contests beyond the two-player case. More realistic extensions of the model can include, for example, heterogeneous players, incomplete information, or embedding the sequential game into a richer dynamic environment.

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24 A Experimental instructions

All amounts in this portion of the experiment will be expressed in points. The exchange rate will be 100 points =$5.00 or 1 point= $0.05. This part of the experiment will have you make decisions over 25 rounds.

Endowment and Expenditure

At the beginning of this part of the experiment you will be randomly assigned to either a Leader or Fol- lower role. You will maintain the same role throughout this part of the experiment. In each round you will be randomly matched with two other participants, and given an endowment of 240 points to expend on a contest. Every group will have 2 leaders who will make their expenditure decisions ﬁrst, which will then be revealed to the Follower, who then decides their expenditure. You can expend any integer number of points from 0 to 240. You will keep any points you choose not to expend. If you win the prize for the round you will receive 240 additional points.

Probability of Winning

After the expenditure decisions are made the sum of expenditures in your group will be calculated. Then the probability of you winning the prize in that round is given by:

Your Expenditure The Sum of Your Group0s Expenditures For example, suppose you chose to expend 20 points and another member of your group chose to expend 30 points, and the third member of your group chose to expend 50 points. Then the probability you will win the prize is:

20 20 1 = = = 20% 20 + 30 + 50 100 5 For another example, suppose you chose to expend 100 points and another member of your group chose to expend 20 points, and the third member of your group chose to expend 40 points. Then the probability you will win the prize is:

100 100 5 = = = 62.5% 100 + 20 + 40 160 8

Payoﬀ in a Given Round

After determining the probability that you win the computer will randomly assign you to a player number. The ﬁrst player will be assigned to the interval from 0 to their probability of winning, player 2 will receive the interval from player 1’s probability of winning to player 1’s probability of winning plus player 2’s probability of winning, player 3 will be assigned the interval from player 1’s probability of winning plus player 2’s probability of winning to 100. The computer then draws a random number between 0 and 100 to determine which member of your group wins the prize. If the number drawn is in your interval, you will win the prize, otherwise another player in your group will win the prize. Using the ﬁrst example from above, if the player numbers are assigned in the order of expenditures then player 1 has the interval from 0 to 20, player 2 has the interval from 20 to 50, and player 3 has the interval from 50 to 100. If the number drawn is less than or equal to 20 player 1 wins, if it is larger than 20 but less than 50 player 2 wins, and if it is greater than 50

25 player 3 wins.

The individual payoﬀ is then calculated as follows:

If you win: If you lose:

240 (Endowment) 240 (Endowment) 240 (Prize) -(Expenditure) -(Expenditure) 240-Expenditure 480-Expenditure

Your earnings

You will compete in a series of 25 rounds, and will be paid your payoﬀ for one of them, chosen at random. Prior to the experiment beginning a random number was drawn and placed in an envelope in front of the class. That number contains the round you will be paid for. At the end of the experiment the round number you were paid for and the payout will be displayed for you to conﬁrm the output. Are there any questions at this time?

We will begin with a non paying practice round to familiarize yourself with the controls. For this round only you can input other players’ decisions and observe your probability of winning the prize.

26 B Additional Tables

n = 2 n = 3 (1) (2) (3) (4) (5) (6) Follower 30.14** 1.313 -5.704 -15.59 7.516 6.390 (12.49) (7.399) (9.211) (9.147) (4.972) (7.459)

Leader 28.42** -2.272 -10.16 -8.439 -3.251 -7.551** (13.04) (6.311) (8.971) (8.885) (2.933) (3.449)

Period -0.996 -0.285 -0.344 -0.291 0.326 0.141 (0.586) (0.339) (0.381) (0.432) (0.204) (0.278)

Wint−1 -11.61*** -18.74*** 4.901 2.068 (3.839) (6.074) (4.438) (4.758)

Follower × Wint−1 9.715 23.17* -38.51** -38.46** (10.45) (11.71) (14.05) (15.62)

Leader × Wint−1 17.51** 23.05** 1.748 7.566 (6.829) (9.131) (6.075) (7.623)

Expendituret−1 0.684*** 0.643*** 0.753*** 0.737*** (0.0443) (0.0508) (0.0341) (0.0420)

Male 5.501 -1.543 (3.621) (3.731)

Econ -4.857 1.728 (3.533) (3.624)

Competitive -1.026 1.813 (2.022) (1.274)

RA 1.089 2.448*** (1.121) (0.840)

Constant 81.42*** 32.20*** 43.83** 85.13*** 11.57* 4.406 (17.26) (9.934) (15.95) (12.48) (5.556) (8.565) N 1188 1188 968 1782 1782 1320

Table 9: Bidding behavior in the last 15 rounds of the experiment. Standard errors clustered at matching group level in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01