Indefinitely repeated contests with incumbency advantage

Abstract

We study an indefinitely repeated Tullock contest in which the stage-game winner gains an incumbency advantage in the next stage-game. The incumbent's advantage allows the incumbent to carry over a proportion of their expenditure in the previous contest to the next contest. Theoretically, this advantage is not predicted to have a large impact on total expenditure. However, in a controlled laboratory experiment, we find the incumbency advantage increases total expenditure by a significant amount. Further, we find that carryover has a discouraging effect on challengers while encouraging incumbents react in a retaliatory manner.

1 Introduction

Many activities can be viewed as contests where each party makes costly expenditures in the hopes of claiming a prize. Given the wide array of situations that can be viewed as contests, it is unsurprising that a large literature has been devoted to the study of such games (see Konrad 2009 and Dechenaux et al. 2015 for surveys). However, the vast majority of the literature focuses on one-shot games, while many situations are better described as dynamically interconnected indefinitely repeated play games. One prominent example is competition between political parties in different election cycles. In such settings, the incumbent may hold an advantage. In this paper, we explore how such an advantage impacts behavior in an indefinitely repeated play contest. Formally, in our setting, a portion of the winner's expenditure in one contest can be carried over to the next contest. In controlled laboratory experiments, we find such carryover increases expenditure for both the incumbent who won in the previous period and the challenger who lost. Further, we find that carryover has a discouraging effect on challengers, while incumbents react in a retaliatory manner.

This paper connects the literature on indefinitely repeated contests with the literature on contests with carryover and incumbency advantage. Brookins et al. (2021) compares indefinitely repeated and finitely repeated Tullock contests with equal expected durations. They find expenditures to be lower in the indefinitely repeated game, suggesting cooperative behavior between the contestants. Brookins et al. (2021) also examines how the discount rate impacts cooperation and reports some evidence that cooperation is stronger when the discount rate is lower, but they do not consider any linkages between contests.¹ While Schmitt et al. (2004) consider a setting with linkages in which both players can carryover investment from one period to the next, they do so in a finite horizon game. Schmitt et al. (2004) find that carryover is sensitive to the decay rate and shifts expenditure to earlier periods, consistent with theoretical predictions. Interestingly, they also find that total expenditure is lower with carryover than without, even though they still observe overbidding as is common in contest experiments. Grossmann et al. (2010) does examine carryover in an infinitely repeated contest, but does so in a setting where there is no incumbency advantage. Using dynamic programming, they derive equilibrium in steady states and further analyze the effects of revenue sharing on competitive balance between the two players. Baik and Lee (2000) also study carryover, but in a two-period model with a contest structure that is very different from that in Schmitt et al. (2004), Grossmann et al. (2010), and our paper. While there are papers that examine incumbency advantage (e.g., Hafer 2006, Polborn 2006, Virág 2009, and Häfner & Nöldeke 2019), these papers have done so in different settings from ours.

2 Theoretical framework

Consider an infinitely repeated Tullock contest with two players who, in each period, compete for a strictly positive prize V by choosing a non-negative expenditure level. We assume that both players are risk neutral and the common discount factor for their future income is $\beta \in (0,1)$ per period. We denote player i's expenditure and his corresponding probability of winning V in period t by $e_{i,t}$ and $p_{i,t}$ , respectively. In our baseline model, we assume the standard Tullock contest success function for both players: $p_{i,t}$ is the ratio of $e_{i,t}$ to the sum of $e_{i,t}$ and $e_{-i,t}$ . Specifically, the probability of winning V in period t for player i is

(1) $\begin{array}{c}{p}_{i,t}=\frac{e_{i,t}}{e_{i,t}+e_{-i,t}}\end{array}$

if $e_{i,t}+e_{-i,t}>0$ , and $p_{i,t}=\frac{1}{2}$ if $e_{i,t}+e_{-i,t}=0$ . Player i's expected payoff in period t is

(2) $\begin{array}{c}{\pi }_{i,t}=p_{i,t}V-e_{i,t}.\end{array}$

The first-order condition for player i in choosing $e_{i,t}$ to maximize ${\pi }_{i,t}$ given $e_{-i,t}$ is

(3) $\begin{array}{c}\frac{\partial p_{i,t}}{\partial e_{i,t}}V=\frac{e_{i,t}}{{(e_{i,t}+e_{-i,t})}^2}V=1,\end{array}$

and therefore, his best response in period t can be written as

(4) $\begin{array}{c}{e}_{i,t}=\sqrt{e_{-i,t}V}-e_{-i,t}.\end{array}$

It follows that there exists a subgame perfect Nash equilibrium supported by the above best response in every period. Specifically, there is a Nash equilibrium in the stage-game where each player chooses the expenditure level of $\frac{V}{4}$ and the expected payoff for each player is $\frac{V}{4}$ each period.²

We now consider how the model changes when the winner of the previous contest (the Incumbent) has an advantage. Specifically, we assume the Incumbent can carry over a fraction of his expenditure from the previous period and combine it with his expenditure in the current period, while the other player (the Challenger) cannot. In this setting, $p_{i,t}$ is also affected by either $e_{i,t-1}$ or $e_{-i,t-1}$ depending on whether it is player i or $-i$ who is the incumbent. Formally, a player is said to be in state I in period t if that player won the contest in period $t-1$ and a player is in state C in period t if the player lost the contest in period $t-1$ . Since the carryover amount is proportional to the Incumbent's previous expenditure, but the Incumbent could have been the Incumbent or the Challenger last period, we index each period of the model by how many consecutive periods the Incumbent has maintained his status denoted by $\tau$ . Thus, $\tau =1$ represents a period in which the current Incumbent was the Challenger in the previous period and $\tau \ge 2$ indicates a period in which the Incumbent was already the Incumbent in the previous period. We let $\tau =0$ denote the most recent period in which the current Incumbent was not the Incumbent. Let $p_{I,\tau }$ and $p_{C,\tau }$ denote the probabilities that the Incumbent and the Challenger win in period $\tau$ , respectively.

We let $u_I\left(e_{C,0}\right)$ and $u_C\left(e_{C,0}\right)$ be the total discounted expected utility of the Incumbent and the Challenger, given $e_{C,0}$ , from period $\tau =1$ onwards, and $v_I\left(e_{I,\tau -1}\right)$ and $v_C\left(e_{I,\tau -1}\right)$ be the total discounted expected utility of the Incumbent and the Challenger, given $e_{I,\tau -1}$ , from period $\tau \ge 2$ onwards. Note that the Incumbent in period $\tau =1$ was the Challenger in period $\tau =0$ so a fraction of $e_{C,0}$ is carried over to period $\tau =1$ , while a fraction of $e_{I,\tau -1}$ is carried over to period $\tau$ for $\tau \ge 2$ , because there is no role change in those periods. Let $\delta \in (0,1)$ be the proportion of expenditure that the contest winner in the previous period can carry over to the current period. The Bellman equation for each player, with a role change in period 1 and without a role change for $\tau \ge 2$ consecutive periods, is given by

(5) $\begin{array}{cc}{u}_I\left(e_{C,0}\right)& =\underset{e_{I,1}}{max}\left(p_{I,1},V,-,e_{I,1},+,\beta ,\left(p_{I,1},v_I,\left(e_{I,1}\right),+,p_{C,1},u_C,\left(e_{C,1}\right)\right)\right)\end{array}$

(6) $\begin{array}{cc}{u}_C\left(e_{C,0}\right)& =\underset{e_{C,1}}{max}\left(p_{C,1},V,-,e_{C,1},+,\beta ,\left(p_{C,1},u_I,\left(e_{C,1}\right),+,p_{I,1},v_C,\left(e_{I,1}\right)\right)\right)\end{array}$

(7) $\begin{array}{cc}{v}_I\left(e_{I,\tau -1}\right)& =\underset{e_{I,\tau }}{max}\left(p_{I,\tau },V,-,e_{I,\tau },+,\beta ,\left(p_{I,\tau },v_I,\left(e_{I,\tau }\right),+,p_{C,\tau },u_C,\left(e_{C,\tau }\right)\right)\right)\end{array}$

(8) $\begin{array}{cc}{v}_C\left(e_{I,\tau -1}\right)& =\underset{e_{C,\tau }}{max}\left(p_{C,\tau },V,-,e_{C,\tau },+,\beta ,\left(p_{C,\tau },u_I,\left(e_{C,\tau }\right),+,p_{I,\tau },v_C,\left(e_{I,\tau }\right)\right)\right),\end{array}$

where

(9) $\begin{array}{ccc}{p}_{I,1}& =\frac{e_{I,1}+\delta e_{C,0}}{e_{I,1}+\delta e_{C,0}+e_{C,1}},& p_{C,1}=\frac{e_{C,1}}{e_{I,1}+\delta e_{C,0}+e_{C,1}}.\end{array}$

and

(10) $\begin{array}{cc}{p}_{I,\tau }& =\frac{e_{I,\tau }+\delta e_{I,\tau -1}}{e_{I,\tau }+\delta e_{I,\tau -1}+e_{C,\tau }},\\ p_{C,\tau }& =\frac{e_{C,\tau }}{e_{I,\tau }+\delta e_{I,\tau -1}+e_{C,\tau }},\end{array}$

for $\tau \ge 2$ . The first-order conditions of the above Bellman equations for the Incumbent and the Challenger in period $\tau \ge 1$ are

(11) $\begin{array}{cc}& \frac{\partial p_{I,\tau }}{\partial e_{I,\tau }}V+\frac{\partial p_{I,\tau }}{\partial e_{I,\tau }}\beta [v_I\left(e_{I,\tau }\right)-u_C\left(e_{C,\tau }\right)]+\beta p_{I,\tau }\frac{\partial v_I\left(e_{I,\tau }\right)}{\partial e_{I,\tau }}=1\end{array}$

(12) $\begin{array}{cc}& \frac{\partial p_{C,\tau }}{\partial e_{C,\tau }}V+\frac{\partial p_{C,\tau }}{\partial e_{C,\tau }}\beta [u_I\left(e_{C,\tau }\right)-v_C\left(e_{I,\tau }\right)]+\beta p_{C,\tau }\frac{\partial u_I\left(e_{C,\tau }\right)}{\partial e_{C,\tau }}=1\end{array}$

with all of the derivative terms in (11) and (12) provided in Appendix A. While the marginal cost of expenditure appears on the right-hand side of each equality, there are three components of the marginal benefit on the left-hand side. The first term is the change in the expected prize amount due to the higher probability of winning in the current period. The second term is the expected return from being an Incumbent rather than the Challenger in the next period due to the higher probability of winning in the current period. The third term is the expected change in the value of winning the contest in the next period due to the carried over amount from the current period.

Without knowing the functional forms of $u_I$ , $u_C$ , $v_I$ , or $v_C$ in (11) and (12), it is not possible to predict the specific path that expenditures will take in this game from an arbitrary initial condition. However, we can derive the differences $v_I-u_C$ and $u_I-v_C$ in steady state where $e_{I,\tau }={\overline{e}}_I$ and $e_{C,\tau }={\overline{e}}_C$ . Specifically, as shown in Appendix A, we can write

(13) $\begin{array}{cc}{v}_I\left({\overline{e}}_I\right)-u_C\left({\overline{e}}_C\right)=& u_I\left({\overline{e}}_C\right)-v_C\left({\overline{e}}_I\right)\\ =& \frac{[({\overline{p}}_I-{\overline{p}}_C)V-({\overline{e}}_I-{\overline{e}}_C)][1+\beta ({\overline{p}}_I-{\overline{p}}_C)]}{1-\beta ({\overline{p}}_I-{\overline{p}}_C)}.\end{array}$

As an example, if $V=1,000$ and $\beta =\frac{3}{5}$ , we find that on the steady-state path, ${\overline{e}}_I=174.2$ and ${\overline{e}}_C=277.4$ . Comparing this outcome to the situation where there is no carryover and each player's investment in the stage-game equilibrium is $\frac{V}{4}=250$ in every period, we find that the Incumbent invests less, because the carryover from the previous period reduces the marginal benefit of the investment in the current period, while the Challenger invests more because of the possibility that $\frac{3}{5}$ of his investment in the current period will be carried over to the next period. If we consider the first period of a repeated contest with winner carryover (i.e., when $\tau =0$ ) when neither player is the Incumbent, it follows immediately that each player will invest more than 250.

3 Experimental design

To explore how winner carryover impacts behavior in an indefinitely repeated contest, we implement a 2 $\times$ 1 between-subject experimental design. For simplicity, both here and in the experiment, we use the term period when referring to the stage-game contest and the term round when referring to the indefinitely repeated super-game.

In the No Carryover baseline, a pair of subjects competed in a series of contests where the probability that contestant i would win the prize $V=1000$ lab dollars in a period was $\frac{e_{i,t}}{e_{i,t}+e_{-i,t}}$ . To help the participants understand the structure of the contest, it was described as a raffle where each contestant could buy tickets at the cost of 1 lab dollar per ticket, and then, one raffle ticket would be drawn at random to determine who received the prize that period. Given the parameter values, the numerical prediction for the No Carryover Baseline is the standard result of $e_i=\frac{V}{4}=250$ and each player has an equal chance of winning the contest and has an expected payoff of $\frac{V}{4}=250$ each period.

In Winner Carryover the player who won the contest in the previous period had an advantage in the current period.³ Specifically, the player who won the previous contest would have $\frac{3}{5}$ of the tickets he purchased in the previous period automatically carried over and entered into the raffle in the current period.

A laboratory session consisted of eight participants in a single treatment. Upon entering the laboratory, subjects were seated at individual computer stations and followed along as a researcher read aloud an initial set of instructions common to all treatments describing a one-shot contest.⁴ Participants then completed four unpaid one-shot practice contests against the computer. Participants were informed that they would play against the computer and that the computer was programmed to purchase a random number of tickets up to 1,000 in each practice contest.

After the practice contests, the subjects again followed along as a researcher read aloud treatment-specific instructions. Participants were informed that half of them had been randomly assigned to the “Crimson” group and the other half had been randomly assigned to the “Gray” group. Participants were informed of their own group assignment and of the fact that group assignments remained fixed throughout the study. Finally, participants were informed they would always be matched with a different person from the other color group each round in such a way that not only did no pair of contestants interact in multiple rounds, no participant's own action in one round could indirectly influence anyone the participant would meet in a subsequent round. This was achieved with a turnpike matching protocol (Cooper et al., 1996).

Once the instructions had been read and any clarifying questions answered, participants proceeded to complete four rounds in the assigned condition. At the end of each period, participants received the following feedback: who won the prize that period, their own earnings for the period, the other contestant's earnings for the period, the number of tickets they purchased, the number of tickets the other contestant purchased, any tickets carried over from the previous period when applicable, and their probability of having won the prize that period. At the end of each of the first three rounds, participants were reminded of the process used to rematch contestants.

To implement the indefinite round horizon with $\beta =\frac{5}{6}$ , the following procedure was used. Before the experiments were conducted, a six-sided die was rolled repeatedly until a six appeared. The number of periods in a round was equal to the number of rolls that had occurred when a six first appeared. This procedure was repeated separately for each round. To maintain consistency across sessions, the duration of rounds in all sessions was based on the same four sequences of die rolls. The actual number of periods in rounds 1 through 4 were 2, 4, 6, and 4, respectively. To allow for behavior to be observed for several periods for each pair, the participants were informed of the process that was used to determine the number of periods in a round, but not the realization. Further, participants were informed that they would complete 12 periods in a round after which it would be revealed if 1) the actual number of periods was 12 or less or 2) the actual number of periods exceeded 12. If the actual number of periods in the round was 12 or less, then after the ${12}^{\mathrm{th}}$ period the participants were informed of the actual duration of the round and all payoffs were based on that duration. If the actual number of periods exceeded 12, then after the ${12}^{\mathrm{th}}$ period, it would be revealed that the round would continue and that it would be revealed after each subsequent period if the round would end.

At the end of the study, one round was randomly selected and subjects were paid based on the randomly selected round (see Azrieli et al., 2020). Lab dollars were converted into $US at the rate of 100 lab dollars = $US 1 and this was common information. To account for the fact that half of the participants would lose money in the first period of a round and that some participants might lose money in several periods, each participant also received an endowment of 1,500 lab dollars. The average salient earnings, including the endowment, were $US 9.66. In addition, participants received a flat payment of $US 5 for completing the one-hour session.

The study was conducted at the University of Alabama's TIDE Lab. Data are reported from 80 participants who completed the study comprising five sessions per treatment.⁵ The participants were drawn from the lab's standing pool of study volunteers who are overwhelmingly business school undergraduates. None of the participants had previously participated in any related studies.

4 Behavioral results

Our data consist of 80 indefinitely repeated contests (rounds) for each treatment and a total of 1920 individual contests (periods). We report the analysis in two subsections. The first examines aggregate behavior between treatments and the latter discusses the behavior of contestants by role and treatment. Throughout the results, claims of statistical significance use $\alpha =0.05$ .

4.1 Aggregate expenditure between treatments

Figure 1 plots the average combined expenditure of both players in a pair by period for each session. The figure suggests that total expenditure is greater in the Winner Carryover treatment than in the No Carryover baseline. Further, expenditures in both treatments appear to exceed their predicted levels from Sect. 2.⁶

Fig. 1 Average total expenditure by a pair in each period. The figure shows average total expenditure by a pair per session each period. Horizontal lines indicate the theoretical predictions of 500 for No Carryover and 451.6 for Winner Carryover. Vertical lines indicate the start of a new round

To statistically test the difference between the two treatments, we estimate the following regression with standard errors clustered at the session level:

(14) $\begin{array}{cc}TotalExpen{d}_i=& {\beta }_0+{\beta }_{1}WinnerCarryove{r}_i+{\beta }_{2}LastTwoRound{s}_i\\ & +{\beta }_{3}WinnerCarryove{r}_i\times LastTwoRound{s}_i+{\epsilon }_i.\end{array}$

The dependent variable is the total expenditure in contest i excluding any carryover, referred to as $TotalExpen{d}_i$ . $WinnerCarryove{r}_i$ is a dummy variable that equals 1 if the observation is from the Winner Carryover treatment and 0 otherwise. To allow for the possibility that behavior changes as subjects gain experience, we include $LastTwoRound{s}_i$ , which is a dummy variable that equals 1 if the observation is from the last two rounds and 0 otherwise. ${\epsilon }_i$ is contest i's unobserved disturbance. Because the first period of each round differs from subsequent periods in that there can be no carryover, we estimate the above regression model using two separate data sets. The results of this regression are presented in Table 1. The estimation in column (1) includes only data from period 1 of each round, while the estimation in column (2) includes data from periods 2 to 12.

Table 1 Regression analysis of treatments and rounds

Variable (coefficient)	(1)	(2)
Constant ( ${\beta }_0$ )	664.0000***	693.6909***
	(21.5677)	(32.9562)
WinnerCarryover ( ${\beta }_1$ )	251.2750***	263.7841**
	(31.7314)	(97.0780)
LastTwoRounds ( ${\beta }_2$ )	$-$ 36.6250	$-$ 50.1182**
	(34.8875)	(20.2573)
WinnerCarryover $\times$ LastTwoRounds ( ${\beta }_3$ )	312.9500***	130.4773
	(68.3053)	(73.9738)
$R^2$	0.3188	0.1468
Number of Observations	160	1760

The dependent variable for each estimation is the total expenditure for each subject pair. Estimation (1) uses only the data from period 1 of each round, while Estimation (2) uses the data from periods 2 to 12. The base category of each regression is the first two rounds of the No Carryover treatment. Standard errors are in parentheses and are clustered at the session level. Significance levels: * $p<0.10$ , ** $p<0.05$ , *** $p<0.01$

From column (1) of Table 1, it is clear that winner carryover leads to higher initial expenditures. The average total expenditure in period 1 is 664.00 in the first two rounds of the No Carryover baseline, which is statistically higher from the theoretical prediction of 500.00, and only nominally decreases by 36.63 in the last two rounds. By contrast, in the Winner Carryover treatment, average total expenditure in period 1 is 915.28 = 664.00 + 251.28 in the first two rounds, which is statistically greater than the average in the No Carryover baseline and statistically greater than the theoretical prediction of 451.60. In the last two rounds, the average total expenditure is 276.32 = $-36.63+312.95$ , which is a statistically significant increase from the first two rounds and statistically greater than the average in the last two round of the baseline.

Now, we turn to the estimation in column (2) for which we use the data from periods 2 to 12. In the No Carryover baseline, the average total expenditure is 693.69 in the first two rounds, which is statistically higher than the theoretical prediction. For the Winner Carryover treatment, in the first two rounds, the average total expenditure is 957.47 = 693.69 + 263.78, which is statistically greater than the theoretical prediction and greater than the average in the baseline. For No Carryover, the average significantly decreases by 50.12 in the last two rounds, but remains significantly above the theoretical prediction. In the Winner Carryover treatment, the average total expenditure increases by 80.36 = $-50.12+130.48$ to 1,037.83, which is statistically greater than the theoretical prediction and the average in the baseline.

4.2 Individual expenditure between roles and treatments

Fig. 2 Histogram of individual expenditure in no carryover. Bins include all expenditures greater than or equal to the the lower bound and less than the upper bound with the exception of the last bin that includes expenditures equal to 1000. The number of observations is 1760. The mean is 334.32 and the standard deviation is 233.26. Data from the first period of each round are omitted

Figure 2 shows the distribution of individual expenditures in the No Carryover baseline. For consistency with the subsequent analysis of the Winner Carryover treatment, this figure excludes data from the first period of each indefinitely repeated contest. There is overbidding with an overall average expenditure that is 133% of the stage-game prediction, but this is not as great as what is typically observed in one-shot Tullock contests (see Dechenaux et al., 2015).⁷ In this regard, our findings are consistent with Brookins et al. (2021), who also found the repeated play reduced expenditure relative to one-shot behavior. However, Brookins et al. (2021) found that repeated play with either a finite horizon or an indefinite horizon with the same expected duration both yield behavior close to the stage-game prediction, whereas we still observe sizeable overbidding.

Next, we consider individual expenditure in the Winner Carryover treatment where we exclude data from the first period of each round as there is no Incumbent and there can be no carryover. Figure 3 plots the distribution of expenditures separately for Incumbents and Challengers. The distribution of expenditures by Incumbents in panel (a) appears to be a rightward shift from the distribution in the baseline (Fig. 2). It also appears that expenditures by Challengers as shown in panel (b) are more dispersed than those by Incumbents from the same treatment and those in the baseline. It is worth noting that these results contrast with Schmitt et al. (2004) who also investigate the effect of carryover. However, in their experiment, there was no advantage to being the Incumbent as both players could carry over expenditures from one period to the next and this likely explains the behavioral difference.

Fig. 3 Histograms of expenditure by Incumbents and challengers in winner carryover. Bins include all expenditures greater than or equal to the the lower bound and less than the upper bound with the exception of the last bin that includes expenditures equal to 1000. The number of observations for each histogram is 880. In panel a, the mean is 498.21 and the standard deviation is 267.23. In panel b, the mean is 499.45 and the standard deviation is 321.73. Data from the first period of each round are omitted

We now consider how behavior in the winner carryover treatment depends not only on role, but on the amount of carryover as well. In the top row of Fig. 4, we present box plots of expenditures by role conditional on the amount the Incumbent spent in the previous period on the x-axis. Thus, carryover in each period is $\delta =\frac{3}{5}$ of this amount. It seems that both Incumbents and Challengers spend more when the carryover is larger. However, it is not clear if the participants are reacting to the carryover per se or simply reacting to the behavior in the previous period. To address this, in the bottom row of Fig. 4, we plot the data from the No Carryover treatment as if it had been from the Winner Carryover treatment to use as reference point for our analysis. That is, we evaluate expenditures by the winner in the previous period and the loser in the previous period conditional on the Incumbent's expenditure in the previous period. From the figure, we find that both Incumbents and Challengers in the baseline spend more when the Incumbent spent more in the previous period, just as in the Winner Carryover treatment. However, the Challengers in the Winner Carryover treatment appear to be somewhat more responsive to the Incumbent's previous expenditure than are Challengers in the baseline.

Fig. 4 Box plots of expenditure in the current period contingent on previous winner's expenditure in the previous period. Bins on the x-axis include all expenditures by the previous winner in the previous period of greater than or equal to the the lower bound and less than the upper bound with the exception of the last bin that includes expenditures equal to 1000. The top portion of the figure shows expenditure of the Incumbents and the challengers in winner carryover. The lower portion of the figure shows expenditure of the previous winners and the previous losers in No Carryover. The number of observations in each plot is 880. Data from the first period of each round are omitted

To statistically evaluate the effects of player i's role and the carryover amount on their expenditure in period t, $Expen{d}_{\mathrm{it}}$ , we employ the following dynamic panel data model:

(15) $\begin{array}{cc}Expen{d}_{\mathrm{it}}=& {\beta }_0+{\beta }_{1}Expen{d}_{it-1}+{\beta }_{2}ExpendOpponen{t}_{it-1}+{\beta }_{3}Incumben{t}_{\mathrm{it}}\\ & +{\beta }_{4}Incumben{t}_{\mathrm{it}}\times Expen{d}_{it-1}+{\beta }_{5}Incumben{t}_{\mathrm{it}}\\ & \times ExpendOpponen{t}_{it-1}+u_i+{\epsilon }_{\mathrm{it}},\end{array}$

where $ExpendOpponen{t}_{it-1}$ is expenditure of player i's opponent in period $t-1$ , $Incumben{t}_{\mathrm{it}}$ is a dummy variable equal to 1 if player i is the Incumbent in period t and 0 otherwise, $u_i$ is player i's individual effect, and ${\epsilon }_{\mathrm{it}}$ is an individual and period specific disturbance which captures residual variations in expenditure for player i in period t.

By construction of the dynamic panel data model in (15), an endogeneity issue arises, because $Expen{d}_{it-1}$ is correlated with the unobserved individual effect $u_i$ . Differencing the above model can eliminate $u_i$ , but there will be another endogeneity issue, because we have $\Delta Expen{d}_{it-1}$ and $\Delta {\epsilon }_{\mathrm{it}}$ on the right-hand side of the differenced model and both are also correlated, because $Expen{d}_{it-1}$ is in $\Delta Expen{d}_{it-1}$ and ${\epsilon }_{it-1}$ is in $\Delta {\epsilon }_{\mathrm{it}}$ . To address this issue, we follow Gill and Prowse (2014) in constructing GMM estimators based on moment equations derived from further lagged levels of the dependent variable and the first-differenced errors (Holtz-Eakin et al., 1988; Arellano & Bond, 1991). Specifically, we use up to six lags of $Expen{d}_{\mathrm{it}}$ as instruments for the differenced equation in the two-step Arellano–Bond estimation to obtain consistent and asymptotically efficient estimators even in the presence of heteroskedasticity.

We conduct the analysis on four separate data sets: (1) data from all rounds of Winner Carryover; (2) data from the last two rounds of Winner Carryover; (3) data from all rounds of No Carryover; and (4) data from the last two rounds of No Carryover. While the first specification is the primary focus of the analysis, the second specification allows for learning in the Winner Carryover treatment, while the third and fourth specifications provide a reference point for the impact of Winner Carryover relative to No Carryover. For each specification, we report the estimated coefficients along with standard errors from the two-step estimates with Windmeijer finite-sample correction (Windmeijer, 2005) in the top section of Table 2. Even though $ExpendOpponen{t}_{it-1}$ and $Incumben{t}_{\mathrm{it}}$ are not endogenous, because they are not correlated with ${\epsilon }_{\mathrm{it}}$ (or even ${\epsilon }_{it-1}$ ), they are weakly exogenous, because they might be correlated with ${\epsilon }_{it-2}$ . Thus, we treat both variables as predetermined in our estimation and include up to three lags of the variables, i.e., how much their opponent invested and the win/loss outcomes in the previous three periods, as instruments for the differenced equation. We also include interaction terms between $Expen{d}_{it-1}$ and $Incumben{t}_{\mathrm{it}}$ , and between $ExpendOpponen{t}_{it-1}$ and $Incumben{t}_{\mathrm{it}}$ as instruments. Finally, we use the generated random number which is used in the experiment to determine the winner of each contest period as an exogenous instrument.⁸

Table 2 Dynamic panel regression analysis of individual expenditure

Variable (coefficient)	Winner Carryover		No Carryover
	(1)	(2)	(3)	(4)
Constant ( ${\beta }_0$ )	500.19***	601.25***	244.11***	222.58***
	(60.92)	(79.70)	(42.65)	(57.59)
$Expen{d}_{it-1}$ ( ${\beta }_1$ )	0.2493***	0.2205**	0.3024***	0.3229***
	(0.0797)	(0.0901)	(0.0849)	(0.1173)
$ExpendOpponen{t}_{it-1}$ ( ${\beta }_2$ )	$-$ 0.1693**	$-$ 0.2674***	0.0037	0.0304
	(0.0708)	(0.0869)	(0.0871)	(0.1043)
$Incumben{t}_{\mathrm{it}}$ ( ${\beta }_3$ )	210.88***	275.19**	181.92***	205.06***
	(67.65)	(119.67)	(44.45)	(56.07)
$Incumben{t}_{\mathrm{it}}\times Expen{d}_{it-1}$ ( ${\beta }_4$ )	$-$ 0.7536***	$-$ 0.8191***	$-$ 0.6541***	$-$ 0.6621***
	(0.0811)	(0.1351)	(0.1131)	(0.1353)
$Incumben{t}_{\mathrm{it}}\times ExpendOpponen{t}_{it-1}$ ( ${\beta }_5$ )	0.2093***	0.2340**	0.0270	$-$ 0.0094
	(0.0757)	(0.1015)	(0.0868)	(0.1038)
Linear combinations of coefficients
${\beta }_1+{\beta }_4$	$-$ 0.5043***	$-$ 0.5686***	$-$ 0.3517***	$-$ 0.3392***
${\beta }_2+{\beta }_5$	0.0400	$-$ 0.0334	0.0307	0.0210
${\beta }_3+{\beta }_4\overline{\mathrm{Expend}}+{\beta }_5\overline{\mathrm{Expend}}$	$-$ 60.91**	$-$ 28.96	$-$ 27.68*	$-$ 10.12
Number of
total observations	1600	800	1600	800
groups	160	80	160	80
observations per group	10	10	10	10
instruments	83	83	83	83
p-value of
Sargan Test	0.4732	0.4303	0.1716	0.4890
Arellano-Bond Test	0.2236	0.1971	0.2009	0.1608

The dependent variable for each specification is $Expen{d}_{\mathrm{it}}$ . The results are based on Arellano–Bond two-step estimation. Data from the first period of each round are omitted. Estimations (2) and (4) use only the data from rounds 3 and 4. Standard errors are in parentheses and are based on the Windmeijer finite-sample correction. Significance levels: * $p<0.10$ , ** $p<0.05$ , *** $p<0.01$ . $\overline{\mathrm{Expend}}$ is an average expenditure per period from period 2 to period 11 based on the sample used for each estimation, i.e., 499.34, 519.82, 334.23, and 320.44 for (1), (2), (3), and (4), respectively

To address the validity of the specifications, we also report the results from two related tests in the bottom section of the table. First, we use the Sargan test to check the validity of the instruments used in the model ( $H_0$ : overidentifying restrictions are valid). Second, we use the robust version of the Arellano–Bond test to determine if the first-differenced error terms are second-order serially correlated ( $H_0$ : zero autocorrelation). Based on the p values reported in the bottom section of Table 2, we do not reject any of the null hypotheses for any specification indicating that the reported analysis is valid.

From column (1) of Table 2, which uses data from all four rounds of the Winner Carryover treatment, we find the marginal effect of a participant's previous expenditure on his current expenditure is 0.2493 for Challenger and $-0.5043=0.2493-0.7536$ for Incumbent and both effects are statistically significant. The estimated difference between the two coefficients is 0.7536, which is also statistically significant. Moreover, we find that the marginal effect of the previous expenditure by a participant's opponent on the participant's current expenditure is $-0.1693$ for the Challenger and 0.0400 = $-0.1693+0.2093$ for the Incumbent. While the marginal effect on the Challenger is statistically significant, the marginal effect on the Incumbent is not. The negative effect on the Challenger can be explained by the discouragement effect on a player in a contest when the other player has a head-start advantage (see Dechenaux et al., 2015). Finally, given an average expenditure from period 2 to period 11 of 499.34, we find that after controlling for the behavior in the previous period under the assumption that both expenditures are equal to each other, and at the average level, Incumbent's expenditure is 60.91 lower than Challenger's expenditure and this difference is statistically significant. If attention is restricted to the last two rounds, the results are largely unchanged as shown in column (2) of the table, although the discouragement effect becomes more pronounced.

For the No Carryover baseline, we create a pseudo-role variable, $Incumben{t}_{\mathrm{it}}$ , to denote if player i in period t won in period $t-1$ . As shown in column (3) of Table 2, the marginal effect of a participant's previous expenditure on his current expenditure is 0.3024 for Challenger and $-0.3517$ = $0.3024-0.6541$ for Incumbent. Both effects are statistically significant. The estimated difference of 0.6541 is also significant despite the lack of actual carryover in the baseline. These values are similar to the ones found in the Carryover treatment suggesting that it is not the carry over per se that is driving the difference in how players respond to their own prior expenditure. However, in contrast to the Winner Carryover treatment, in the No Carryover baseline, the opponent's previous expenditure does not have a significant impact on the Challenger's expenditure. Thus, there is no evidence of the discouragement effect in the baseline. Moreover, given an average expenditure of 334.23, we find that after controlling for the behavior in the previous period, Incumbent's expenditure is on average only 27.68 lower than Challenger's expenditure and this difference is less pronounced that what we observe in the Winner Carryover treatment. Finally, the results in column (4) are similar to those in column (3) indicating that behavior does not change with experience.

In summation, we find that Carryover leads to greater expenditures. Incumbents and Challengers respond differently to the behavior of the previous contest. Incumbents’ expenditures decreasing with their own previous expenditure similarly in both treatments. In contrast, Challengers’ expenditures increase in their own previous expenditure in both treatments. We do not find any effect of the previous expenditure by the opponent on the current expenditure of the Incumbent or the Challenger in the No Carryover baseline. However, in the Winner Carryover treatment, we find a positive effect of the opponent's expenditure on the Incumbent and a negative effect on the Challenger. This suggest that an aggressive action by the Challenger will lead to retaliation by the Incumbent, but a similar action by the Incumbent will discourage the Challenger.

5 Conclusion

Because many naturally occurring situations, like political campaigns, can be modeled as contests, an extensive theoretical and behavioral literature has developed on this topic. Scholars have recently begun considering the effects of repeated interactions in contests as part of the growing interest in repeated play games more generally. This paper contributes to that literature as we consider indefinitely repeated play Tullock contests with dynamic linkages between stage games. Specifically, we consider the effects of the winner in one period being able to carry forward a portion of their expenditure to the next period as such an incumbency advantage is likely to accrue in settings like political campaigns.

In a series of controlled laboratory experiments, we find that winner carryover substantially increases total expenditure in comparison to the indefinitely repeated game with no carryover. Further, in both cases, the observed level of expenditure far exceeds the theoretical predictions. Such a pattern is consistent with popular press accounts of large spending by political campaigns. We find that Incumbents and Challengers respond differently to the behavior of the previous contest when deciding how much to expend in the current contest. Incumbents’ expenditures are decreasing with their own previous expenditure similarly in both treatments. This suggests that even though players with an incumbency advantage reduce their expenditure, this is not due to the carryover per se, but rather a behavioral reaction to having won the previous contest. By contrast, Challengers’ expenditures increase in their own previous expenditure in both treatments. In the No Carryover baseline, we do not find any effect of the previous expenditure by the opponent on the current expenditure of the Incumbent or the Challenger. However, in the Winner Carryover treatment, we find a positive effect of the opponent's expenditure on the Incumbent and a negative effect of the opponent's expenditure on the Challenger. This suggest that an aggressive action by the Challenger will lead to retaliation by the Incumbent, but a similar action by the Incumbent will discourage the Challenger.

While our experiment is meant to mimic naturally occurring settings like political campaigns, there are a myraid of differences between the two. For example, in practice, the Challenger may receive some carryover benefit, even if it smaller than the benefit to the Incumbent. In such a setting, the effect of carry over may be smaller than what we observe. Further, the benefit to the incumbent may depend on the duration of the incumbency, while we modeled the benefit extending to only one future contest. We hope that our study spurs further research on these issues and on indefinitely repeated games with intercontest linkages more generally.