2 Literature review

One of the most popular solutions applied in the regulatory policy of pension funds industry has been to establish fee caps. Skypala (Reference Skypala2015) provides a meaningful illustration from the UK, where fees have risen since caps were discarded in 1980. This implies that economies of scale, which has been confirmed by many studies of the pension fund industry (e.g., Dobronogov and Murthi, Reference Dobronogov and Murthi2005; James, Reference James2005; Sy and Liu, Reference Sy and Liu2010), did not translate into more favourable conditions for clients but only into bigger companies’ gains (Rudolph and Rocha, Reference Rudolph and Rocha2007). In fact, in many countries with legal caps, the market average coincided with the legal maximum fees, especially in the mandatory systems (Ionescu and Robles, Reference Ionescu and Robles2014). Consequently, caps should not be viewed as perfect tools. Additionally, doubts about the ‘right’ level of caps will always be present (Whitehouse, Reference Whitehouse2000).

A unique solution was employed in Sweden, where the government introduced the PPM (Prempensionsmyndigheten) – or Premium Pension Authority in English. PPM is a clearinghouse for fund transactions, which keeps individual accounts, collects and provides (daily) information on participating funds, provides information services to participants and is the monopoly annuity provider (Palmer, Sweden's New FDC Pension System, Reference Palmer2005). The collectively provided accounts’ administration services drive down the actual costs for fund managers and they must rebate to PPM a share of the fees, which are further passed by PPM to investors (Sundén, Reference Sundén2004). PPM can also execute its monopolistic position by requiring that the marginal cost of investing additional funds would decrease with the volume of PPM assets invested (Palmer, Reference Palmer2008). Additionally, charges are expressed as a percentage of PPM assets held by all the PPM funds owned by the managing company, which additionally supports the reduction of commissions.

Of course, one could argue that instead of employing the Swedish model or complex tools, like the reverse auctions advocated in this paper, the regulator could simply lower the fee caps, which would force all MFs to lower their fees. One argument against this is simple: due to the asymmetry of information, the regulator does not have perfect knowledge on how deeply it can cut the maximum fee without endangering the financial stability of MFs and the efficient functioning of the market. Any non-market intervention of that type is risky, as it might cause more problems than it solves. Auctions, as a market institution, are believed to be generally more efficient with respect to this problem,Footnote ¹ although their relative efficiency depends on market circumstances and therefore should be experimentally tested.

Nevertheless, it is intriguing to consider why market competition has not worked. The Bertrand competition model demonstrates that even with just two firms in the market, one can expect a price war. One reason for the lack of price competition in the pension fund industry could then simply be that the market does not meet the assumptions of the Bertrand model: it can be neither homogeneous nor perfectly transparent, and additionally, members may face exit barriersFootnote ² – all of which make a reduction in commissions not very profitable for a fund. Cross-country experience reveals that clients do not pay much attention to commission levels. Instead, they are mainly driven in their choices by marketing activities (advertising, promotional gifts) (Queisser, Reference Queisser1998; Chybalski, Reference Chybalski2011), which have frequently constituted a significant part of MF expenditures (Impavido and Rocha, Reference Impavido and Rocha2006). Moreover, the complexity of fees charged and the presumably low financial literacy of many clients may weaken the impact of information about commission cuts, thus additionally lowering the expected gains of such a decision by an MF.

However, even with a homogeneous and transparent market with no entry/exit barriers, price competition is not necessarily inevitable. Sweezy's kinked demand curve model of oligopoly predicts that firms would be reluctant to lower prices, as they would fear that it would be matched by a similar cut in prices by competitors. Thus, a reluctance to lower commissions might be seen as reflecting a tacit agreement among MFs. Simply, all MFs understand that cutting commissions could be a spark that would start a price war, which they want to avoid. If the deteriorating market situation or regulatory shifts do not force them to make changes, they can all stick to their initial prices for a longer period of time.

This latter model may suggest that some additional tools should be utilized to induce market competition. One potential solution may be a reverse auction mechanism. In fact, some countries have already used this idea in their pension systems. Bolivia ran an international competitive bidding process to choose MFs allowed to operate in its market (von Gersdorff, Reference von Gersdorff1997). A similar move was later undertaken by Kosovo, a step recognized as the main cause of much lower fees experienced in both countries compared with other economies in their regions (James, Reference James2005). India introduced an auction in 2009 to determine the lowest cost MF. As a consequence of using an unorthodox auction mechanism, the management fee settled at close to zero.Footnote ³

Other evidence that has inspired our study comes from Chile and Peru. In 2008, the Chilean government introduced a reverse auction designed to force operating MFs to lower their fee levels. In the auction, which is held every second year, MFs are asked what level of fees they would apply to new contributions. The winner, the MF that indicates the lowest price, is rewarded with the right to acquire the contributions of all new participants joining the pension system for a period of 2 years (when the next auction will be conducted). Moreover, the benefits of the lower fees are experienced by all PF participants, whose contributions are managed by the winning company, as price discrimination is not allowed so that the MF must charge all its clients the same fee. Apart from the benefit of acquiring new clients, the winning MF gains public attention and can expect additional clients who switch from MFs that charge higher fees. Figure 1 demonstrates that the years after an auction brought continuing decreases in fee levels charged.

Source: Calculations based on statistical data from Centro de Estadisticas de Superintedencia de Pensiones, http://www.safp.cl. All data are for the 31.12 of the respective years. Average values are weighted by the number of fund members. The average fee level for 2014 is estimated from (Ionescu and Robles, Reference Ionescu and Robles2014).

Figure 1. Changes in fee levels charged by Chilean pension funds after the introduction of the auction mechanism.

The 2012 Peruvian pension reform introduced an auction system very similar to the one successfully implemented in Chile, with the same reward system, involving the acquisition of all new clients for the following 2-year period. The reform in Peru changed the system of fees, converting from a system of fees based on salaries to a system of fees based on account balances. As all new clients will pay fees based on account balances, existing clients will be charged a mixed commission partly based on new contributions and partly based on account balances. The transition period will last until 2023 when the contribution fee disappears.

The experiences of Chile and Peru suggest that the introduction of auctions might be an efficient means of reducing fees charged by MFs. Nevertheless, before implementing a similar concept in the other market, a number of issues must be reconsidered. Up to our best knowledge, in the existing literature, there are no studies discussing the selection of the auction mechanism in this particular area. At the same time, an efficient auction design always depends on the properties of the given market and the specific goals of the auctioneer. Therefore, our research objective is to deliver a framework for setting up the laboratory experiment, which addresses these key questions. In this study, we hope to bridge this gap.

3 The pension system and MF market

To verify the hypotheses we need to establish a set of assumptions referring to the pension system and MFs market. In our model, we have decided to employ a few key points from Polish pension reality. This is due to the fact, that Poland, similarly to many Central and Eastern European (CEE) states (Schwarz, Reference Schwarz2011), is an example of a country which has introduced the mandatory capital pension pillar according to the policy recommendations described in the seminal (World Bank, 1994) report: Averting the Old Age Crisis. Consequently, in this section, we sketch major elements of Polish system and underline the regional commonalities to justify the further selection of model assumptions.

4 The model

The main hypothesis proposed in this study is that the introduction of an auction will lead to a substantial decrease in management fees, that is, the average management fee after the auction will be significantly lower than in the periods prior to it. Apart from the main hypothesis, there are additional specific hypotheses that concern individual decisions made by auction participants or the efficiency of various auction mechanisms. These hypotheses will be introduced in detail in the upcoming chapters.

Before we describe the experimental protocol, as well as the results, we need to explain the assumptions that were made in modelling the simulated market. First, we have decided to cut the number of MFs to 6, compared with 12 in the actual Polish market. This change was motivated by the feasibility of the project; with 12 students simulating one market the cost of the whole research would be too high. Even though the simulated market did not have to mimic the real one, as our goal was to compare the relative efficiency of auctions and not to make detailed forecasts, we tried to keep the concentration level in the experiment as close as possible to the real one. That was done by minimizing the difference between the real market HHI and the modelled one, using two HHI measures: the standard one, and the normalized one, where the latter was calculated according to the following formula:

(1)$$nHHI = \displaystyle{{HHI - (1/n)} \over {1 - (1/n)}},$$

that is, normalized for the number of firms. This led us to a market structure with the following market shares: 29%, 26%, 20%, 10%, 10%, and 5%.Footnote ¹⁰

It was assumed that costs of an MF have the following linear form:

(2)$$TC(A) = \alpha A + \beta, $$

where TC is the total cost, A is the value of net assets, and α, β are parameters.

Although the linear form of the total cost function and the assumption that the parameters, α and β, have the same values for all MFs are simplifications needed for experimental purposes, it turns out that the specification fits the data extremely well, showing that there are grounds for believing that variable costs depend linearly on asset value. The parameters of Equation (2) were estimated using the linear regression and data from the Polish MFs market at the end of 2015,Footnote ¹¹ resulting in the values α = 6.01 PLN for each 1,000 PLN of net assets and β = 10.86 mln PLN.Footnote ¹²

MFs make decisions concerning four variables: contribution fee (f _C), management fee (f _m), and investment levels in two risky assets (r ₁, r ₂). The contribution and management fees cannot exceed the current caps, which are 1.75% and 0.54% (annually), respectively.

The investment decision is made as simple as possible. Participants decide only on the fraction of the money to be invested in two types of risky assets, where r ₁ is the amount of money invested in medium risk assets, and r ₂ is the amount of money invested in high-risk assets. The rest is invested in risk-free bonds. Participants know the distributions of returns from both types of investment, which are modelled to have independent normal distributions.

According to the aforementioned default option of automatic enrolment in the 1^st pillar, introduced in 2014, only approximately 15% of PF membersFootnote ¹³ decided to continue transferring part of their pension contributions to the 2^nd pillar. Henceforth, we will call these members the active members and the remaining majority the inactive members (their accumulated funds are still managed by the MF, but no new contributions are made). When weighted by the value of assets, the share of active members is actually higher, and closer to 25% and so in the model, we assume it to be on this level. As only a very small percentage (approximately 1%) of new peopleFootnote ¹⁴ entering the system now decide to become members of PFs, in the model we assumed that there are no new inflows of members to the system.

The marginal analysis of the MF's profit function showsFootnote ¹⁵ that a decrease in the management fee by 0.01 percentage point will be profitable if it is compensated by the acquisition of additional 68,100 members. In case of the contribution fee, this number equals 1,300. Taking into account the actual shifts in the number of members observed in the Polish market, and the fact that the aforementioned threshold values are underestimated,Footnote ¹⁶ it is never in MF's interest to cut the management fee. In contrast to the management fee, though, one should expect a greater willingness to lower the contribution fee on the MF side. This is due to the fact that new contributions do not carry much weight in the MF revenue structure and so obtaining the new members (with their accumulated assets) might be tempting. Decisions taken by MFs in the Polish market are in line with the conclusions reached in the marginal analysis: three MFs have cut their contribution fees, and none have ever cut their management fees. The profit analysis demonstrates that it is not in an MF's interest to lower the management fee, which is why we argue for the introduction of an auction. Although this research was inspired by the Chilean and Peruvian experiences, the modelled market differs so much from the South American ones that it demands a completely different approach. First, the market concentration level is lower than that in the South American countries. Second, the Chilean auctions concerned only the fee level on new contributions, as that was the main source of revenues for MFs. As a consequence, the natural reward in the auction was the acquisition of new members. This solution created incentives for an MF to lower its fee on new contributions and to benefit new members by allocating them to the lowest cost funds. In the modelled market we have two types of fees, and it is the management fee that weighs more heavily on members. Moreover, due to recent trends observed in the CEE states, the flow of new contributions has been significantly reduced. Therefore, we modelled the number of new members entering the system to be very small, and so a reward system like that of Chile would most likely not play its allotted role in the markets like Polish one.

Thus, we suggest a different reward mechanism – that the winning MF be rewarded with the acquisition of some of the assets of already-existing members. For that purpose, the assets of inactive members could be used.Footnote ¹⁷ Inactive members do not pay a fee on contributions (as their new contributions are transferred to the 1^st pillar only), so the fee on new contributions cannot be a criterion in the auction. Those members would only be interested in the management fee and the rate of return on their assets, so those two variables would be the criteria used in the auction.

In consequence, it was decided to experimentally test a two-criteria scoring auction, with the main criterion being the management fee and the secondary criterion being the rate of return. The rate of return is not a variable decided upon by auction participants, as its value is calculated as the actual rate of return achieved by the fund in the periods following an auction. Each fund learns about those values and points reached for this criterion prior to the auction, so it knows where it stands relative to competitors.

It is difficult to determine how the two criteria should be weighted in the scoring auction.Footnote ¹⁸ The rate of return might have a tremendous impact on the situations of fund members: huge profits can double the value of the fund, whereas substantial losses can consume a major part of it. The role of the management fee is less spectacular; the values it could take range between 0% (hypothetically) and 0.54% (maximum), so it would never dramatically affect the situations of fund members.Footnote ¹⁹

On the other hand, the value of the management fee is stable. The clients of the winning MF are guaranteed that they will not be charged a fee higher than the declared commission for a period of 2 years. In the case of the rate of return, there is no guarantee; hence, persistence in rates of return among PFs has not been observed (Kominek, Reference Kominek2006).

Taking that into account, we decided to assign a much higher weight of 0.8 to the management fee and only 0.2 to the rate of return. With the maximal score set at 1,000 points, an MF can maximally reach 200 points for the rate return and 800 points for the management fee. In the case of the management fee, a linear scoring rule is used, with a maximal value of 0.54% and a minimal value of 0.3%,Footnote ²⁰ that is, the score is calculated using the following formula:

(3)$$S_m = S(\,f_m ) = 800 \times \displaystyle{{0.54\% - f_m} \over {0.24\%}}, $$

In the case of the rate of return, a linear scoring rule would be difficult to introduce, as it is hard to predict what range actual rates of returns would take, and we would like to avoid setting them arbitrarily for each new auction. For that reason, it was decided to introduce the highest bid – lowest bid scoring rule (Dimitri et al., Reference Dimitri, Piga and Spagnolo2006):

(4)$$S_r = S(r) = 200 \times \displaystyle{{r - r_{lowest}} \over {r_{highest} - r_{lowest}}}, $$

where r _lowest, and r _highest are, respectively, the lowest and highest rates of return obtained by all MFs participating in the auction.

The main difference between formulas (3) and (4) is that in the former case, the threshold values are predetermined, whereas in the latter case, they result from the specific data. The consequence of this is that the MF with the lowest rate of return will earn 0 points on that criterion, and the MF with the highest rate of return will earn 200 points. The maximum 200 point differential in the case of the rate of return corresponds to a 0.06% differential in the case of the management fee, that is, by how much the poorest performing MF would have to cut its management fee to make up for lost points.

Of course, we could consider a different auction design, for example, one in which MFs declare a minimal guaranteed rate of return over the next 2 years. In the case of this auction protocol, we could consider a much higher weight for the rate of return. However, there are some risks in implementing a solution of this type. Will MFs be willing to risk declaring any positive value in such an auction? And if so, would that not result in a much safer investment policy in the coming years, providing members of the winning fund with the lowest rate of return in the market? Finally, there is an adverse selection risk. There is a danger that the MFs in the worst financial conditions would be desperate enough to risk guaranteeing positive rates of return, which could have fatal consequences in the future for both these MFs and the financial security of their clients.

A decision to use a scoring auction does not predetermine which auction mechanism is adapted. As a benchmark model, a first-score auction was the first auction mechanism we applied. Under this mechanism, participants make sealed-bids of management fees, and the winner is the MF that obtains the highest score in terms of both criteria employed. After the auction, the winning bidder applies the declared value of the management fee, which cannot be lowered for 2 years.

However, there is one flaw that arises in the application of this or any other standard auction rule. As the auction forces the winning MF to lower the management fee, it does not introduce any incentives for losing MFs to do the same. Of course, this problem has also been observed with the South American auctions described in Section 3: as Figure 1 demonstrates, the minimal fee level applied is much lower than the weighted average; the largest MFs did not necessarily cut their fees to such an extent. When planning the auction in Chile, the hope was that other MFs would be forced to lower their fees as well in response to greater market pressure and potential shifts of their members to the lowest fee fund. In Poland, due to the non-transparency of the market and the limited number of active members left, this problem might be even more serious.

Therefore, we decided to experimentally test a non-standard auction mechanism that is widely known in the auction literature, namely, a (first-price) all-pay auction (Krishna, Reference Krishna2002). This is a sealed-bid mechanism, just like the first-score auction, with one important difference – under this mechanism, it is not just the winner that has to pay his or her bid; the rule applies to every bid made in the auction. If the all-pay auction design was introduced in the case of the scoring auction under study, all participating MFs would have to maintain the management fees asked in the auction for a period of 2 years, even if they lose the auction. The auction theory does not provide an unequivocal conclusion on the efficiency of all-pay auctions, given risk-averse bidders and market asymmetries (Baye et al., Reference Baye, Kovenock and De Vries1996). Thus, its relative efficiency compared with the first-price sealed-bid auction must be tested experimentally.

5 The experimental design

The experiment was conducted as a multi-round experiment, with an unknown number of rounds and a rule that one round corresponds to 1 year in the real economy. It was designed and implemented using the zTree environment (Fischbacher, Reference Fischbacher2007). Students of the Wroclaw University of Economics voluntarily enrolled in the experiment through the Internet.Footnote ²¹ Although there were students from various years of study, who therefore had different levels of economic knowledge and experience, we did not observe any significant differences in their decisions.Footnote ²² The experiment began with instructions that included true or false questions to check the students’ comprehension of the game rules. Following the instructions, a one-trial round was played, and all doubts concerning the rules were publicly cleared up before starting the actual experiment.

Significant parts of the instructions and explanations were intended to ensure that all students understood the incentive system. As the model was asymmetrical, with students differing in the starting numbers of assets and profits achieved, it was decided that the participants’ rewards would be based on their relative performance, that is, on the extent to which they were able to increase their MF's profits. Therefore, the final reward depended on the relationship between the average profit achieved by a participant and his or her starting profit, determined by the initial game parameters.

Students could keep track of their rewards, as information about their current payoff levels was available on computer screens throughout the game. It was emphasized on several occasions that each participant's reward depends only on his or her profits and does not depend on the profits amassed by other MFs.

As an incentive system extra points for the classes, which affected students’ final grades, were awarded. There are reasons why this type of incentive was believed to work best. First, when using classroom points, we avoid any income effects, which are known as a weak point in financial rewards. All students were in a similar position: the experiments were carried out at the beginning of the semester when none of the students had any points from their courses and therefore, all students reached similar utility from each awarded point. The maximum number of points that students could win during the experiment was 10% of the total number of points from the course, which is a significant reward for any student. Our experience from the previous years, concerning auction experiments with classroom points, demonstrates that this is, in fact, a very strong incentive: the number of volunteers was spectacular, participants were highly motivated, and when asked in a post-experiment survey to value one classroom point, the median answer was 30 PLN. Given an average payoff of 3.3 points, the average payoff was valuated at 100 PLN, or 23 EUR,Footnote ²³ a significant value in the case of Polish students (Kuśmierczyk, Reference Kuśmierczyk2013). Therefore, we can say that our experience demonstrates a known fact, noted by Friedman and Sunder (Reference Friedman and Sunder1994, p. 43.): ‘grades can elicit high levels of motivation and effort from subjects without spending money’. Finally, a system of financial rewards would entail tax reporting obligations by participants, an additional obstacle that could discourage some potential volunteers.

After the instructions and the trial round, the actual experiment began. At first, students were not informed that an auction would be introduced into this market, as we needed to observe how competition would work without it. After two rounds of competition, students were informed about the auction and provided with instructions. Thereafter, the auction was run every second year.

Each decision round was followed by a results window, which provided all the necessary information:

• • on the financial results of the individual MFs: revenues from all sources, costs, the rate of return, profit, number of members, and the current payoff;

• • on the current market situation: fee levels and fee level changes for all MFs, rates of return achieved, shifts in the number of members, and profit changes.

Apart from the abovementioned elements, participants were allowed to use an online text messenger. Chat messages were sent to all market participants at the same time, and there was a limit of two messages per round. The chat window was introduced to facilitate potential collusion that could occur in this market. Although students were encouraged to use it freely, in fact, the chat did not play a significant role in the experiment.

All experiments were conducted using the Wroclaw University of Economics’ laboratory facilities. The average time of an experiment was 2 hours, which allowed us to simulate 9 years in the market.

5.1 Game parameters/treatments

Summing up, the simulation of the pension fund market was run with the following assumptions:

1. A. Six MFs compete in the market, with the following market shares: 29%, 26%, 20%, 10%, 10%, and 5%.

2. B. The total number of members is 16,500,000; 75% of them (i.e. 12,375,000) are inactive, and 25% (i.e. 4,125,000) are active. These numbers remain constant throughout the simulation: there are no new entrants or withdrawals from the system.

3. C. The members are homogeneous. As the total value of assets of all PFs is 140 billion PLN, the value of assets per member is 8,485 PLN. All active members pay a new contribution each year of 650 PLN.Footnote ²⁴

4. D. All MFs have cost functions given by (2), with α = 3 PLN/1,000 of assets and β = 1,000,000 PLN.Footnote ²⁵

5. E. The maximal values of the fees match the real ones, that is, 0.54% annually in the case of the management fee and 1.75% in the case of the distribution fee. In the game, the players start with values 0.50% and 1.50% for the management fee and distribution fee, respectively, so as not to suggest to them that the maximal fees are necessarily optimal.

6. F. Apart from the decisions about fee values, participants determine what shares of assets are invested in three types of investments. The rate of return on risky investments is drawn from a normal distribution, $N(7.58\%, \;5.72\% )$, and for moderately risky investments, the following normal distribution is used: $N(4.29\%, \;1.92\% )$. The remaining assets are invested in a risk-free investment, with a rate of return of 2%. The randomization processes for the two risky investments are independent, as are the draws in each subsequent period. However, the draws for different members in the same decision period are not independent; in fact, the returns on risky investments of various MFs are very close to each other (they differ by random, marginal values to differentiate the final rates of return, which affect the premium fees and the auction results).

7. G. The shifts of members between the funds were calculated using a simple algorithm. It was assumed that the members might decide to change funds due to one of the three criteria: the management fee, the distribution fee, and the last round's rate of return. For each of these criteria, the maximum number of members changing funds was independently set at 1% of all active members, and potential switches of members were distributed proportionally according to the parameter values. Setting the values on these levels made a decrease of the management fee unprofitable, as determined by the theoretical analysis.Footnote ²⁶ In the period just after the auction, the proportion of members considering a change of fund was increased to 5% to simulate an effect of greater awareness on the clients’ side.

8. H. With respect to the decisions periods, various pieces of information were revealed to facilitate the decision process: the expected rate of return, the expected number of members who would shift, the expected profit. Students could test all of them for various fee levels and the investment options considered. Forecasts were made with the ceteris paribus assumption, that is, none of the competitors change any of their parameters.

9. I. To help the students make rational decisions concerning bids in the auction periods, they were given information about the estimated break-even value of the management fee, calculated from the following relationship:

(5)$$E\left[ {TP(\,f_m^0 \vert {\rm ^{\prime}}lose{\rm ^{\prime}})} \right] = E\left[ {TP(\,f_m^{^\ast} \vert {\rm ^{\prime}}win{\rm ^{\prime}})} \right],$$

where $f_m^* $ is the break-even value of the management fee, understood as the value of the management fee that, after an MF has won an auction (and gained additional members), yields the MF the same profit it would have had, if it had lost the auction and lost 10% of its inactive members but retained a higher (current) management fee ($f_m^0 $). The students received exhaustive explanations of how this value should be interpreted.Footnote ²⁷ Naturally, the break-even management fee values depended on the fund's market share, as it was more profitable for small funds to cut this fee. Table 1 presents the values of $f_m^{\ast} $ at the initial market shares as well as the point advantage (ΔS _m) in the auction resulting from the scoring system used.

Table 1. The values of the break-even management fee levels at the initial market shares

Source: Own study.

As Table 1 demonstrates, the auction was, in fact, asymmetrical, with an a priori advantage given to the smallest MF. The additional factor that introduces asymmetry into this market was the points received for the accumulated rate of return, which were revealed to participants prior to the auction.

Apart from multi-criteria auctions, which are our main focus of interest, we decided to run additional experiments that would test the efficiency of a one-criterion auction, that is, the auction in which the management fee is the only criterion by which the winner is established. The elimination of the accumulated rate of return from the criteria list has two important consequences, from the perspective of auction theory.

First, it makes the auction less asymmetric. The smallest funds still have an advantage, as they have a lower break-even fee, but because points for the rate of return are no longer added, no MF can be a priori sure of winning the auction.

Second, in a one-criterion auction, we no longer need the scoring function, as the winner is simply the MF that asks the lowest management fee. That means that the minimal value of the management fee (for two-criteria auctions, it is 0.30%) is no longer needed. It is, however, hard to predict the consequences of its elimination: on the one hand, MFs are free to offer lower fees in the auction; on the other hand, there is no natural anchor value in the game, and hence, firms might offer higher fees.

The main factor changed in subsequent treatments of the experiment was the auction mechanism. Table 2 presents information on all treatments used in the experiment.

Table 2. The experimental treatments

Source: Own study.

All treatments started with two decision periods, without any information or even a hint that any auction was scheduled. As those periods are identical in all treatments, the decisions made were grouped together and are referred to as the NA (i.e., non-auction) treatment. In the analyses that follow, when referring to the auction treatments, we are usually comparing the results of the seven periods that followed the first introduction of an auction. Nevertheless, in a few cases, it will be helpful to use the information from all nine decision periods. To avoid unnecessary confusion, we will use the same symbols in both cases, with an additional note if needed.

6 Empirical findings

In this part, we start by looking at the main hypothesis, that is, whether the auctions successfully lower management fees, which is the main rationale for their potential implementation. Additionally, we will compare the efficiency of the auction mechanisms under study. A detailed analysis of the participants’ strategies is presented in Appendix B.

The first observation, based on these data, is a confirmation of the prediction that with no auction, a decrease in the management fee should not be expected. The simulations were started at a value 0.5%, so the participants had to decide which was more profitable: to decrease or increase its value. The average value of the management fee in the first two periods (i.e., prior to the auction) was 0.5152%, demonstrating a shift upward, closer to the maximal value of 0.54%. In the case of the distribution fee, shifts in value were observed in both directions, and the average value is very close to the starting one.

Auctions did generally play their role in bringing down management fees. In all cases under study, both the average and minimal values of this fee were significantly lower than before the auction. However, one of the auction mechanisms turned out to be much more efficient than the others – the two-criteria all-pay auction forced participants to cut their fees most sharply. One reason for this could be that the minimal value of 0.3% served as an anchor. A management fee that low has been observed on several occasions under treatment 2ALL but was very rare under treatment 1ALL.

To test the statistical significance of the management fee values, a series of statistical tests was run. Before reporting the results of those tests, however, we must look more closely at the data collected. The statistics presented in Table 3 were calculated on the basis of individual decisions taken by participating students. With seven sessions conducted in each treatment, the total number of individual decisions is 7, quite a high number. For example, under treatment 2FSA, 504 individual decisions would be taken (see Table 2). However, the main problem with testing these individual decisions is their interdependence: it would be naive to believe that students’ decisions concerning management fee levels in subsequent rounds were independent (in fact, strong dependency among those decisions was observed).

Table 3. Fee levels prior to and after the auction (in % values)

Source: Own study.

Note: Weighted averages are the fee average levels weighted in all decision periods by market shares. The values in brackets are standard deviations.

Therefore we have decided to run the statistical tests for three groups of aggregated data:

1. (a) Stud data, that is, students’ averages; for each participant just one number is taken into account, that is, the average of all decisions, which reduces the number of data to N _i; the dependency problem is avoided, but the disadvantage is that we are unable to analyze the weighted averages, that are of most concern;

2. (b) Mark data, that is, market averages: taking the weighted average management fee for each session, reducing the number of data to 7G _i; the benefit is working on weighted averages, but the disadvantage is the potential problem of data dependencies;

3. (c) Group data, that is, group averages; we take just one number for each experimental group: the sessions’ average value of the market-weighted average management fees; this avoids the dependency problem, and enables us to work on weighted averages, but reduces the size of a data sample to just G _i.

The reduction in the number of data could be a huge problem, as decisions concerning management fees do not follow a normal distribution; in fact, such data are not even symmetric. This necessitates the utilization of rank sum tests, which might be inconclusive, given the limited amount of data. All test results reported below were obtained using the one-tailed Wilcoxon–Mann–Whitney test (Hollander and Wolfe, Reference Hollander and Wolfe1973) in R.Footnote ²⁸ The null hypothesis is that the measures of central tendency are equal; therefore, rejection of the null hypothesis indicates that differences between the statistics are statistically significant.

The paper's main hypothesis is that auctions help reduce the average management fee. Therefore, first, the results of treatments 2FSA, 2ALL, and 1ALL were tested against the results of the NA treatment, that is, against data from the first two sessions, run prior to the auctions. Second, we pairwise tested the hypotheses that the management fees in treatments 2FSA, 2ALL, and 1ALL are equal, with the alternative hypotheses that $f_m^{2ALL} \lt f_m^{1ALL} \lt f_m^{2FSA} $. Table 4 provides the p-values.

Table 4. P-values for Wilcoxon–Mann–Whitney test concerning the management fee values

Source: Own study, using R.

As can be seen, all test results are conclusive and strongly support all hypotheses. The management fees from the post-auction rounds are all significantly lower than those from the prior-auction rounds. The management fee level reached in the two-criteria all-pay auction (treatment 2ALL) is significantly lower than that reached in the one-criterion all-pay auction (treatment 1ALL), which in turn is significantly lower than that reached in the two-criteria first-score auction (treatment 2FSA). Moreover, the test leads to a rejection of the null hypotheses even in the case of the highly reduced group data, which avoid the potential problem of data dependencies.

Figure 2 compares the average levels of the management fees in all treatments, showing that the higher efficiency of the multi-criteria all-pay auction is not the effect of the shock resulting from the introduction of the new, non-standard auction rule but is observed in all periods of the experiment.

Source: Own study.Note: The graph includes the first two periods, which took place prior to the auction.

Figure 2. The average level of the management fee by rounds.

More insight into potential differences in auctions’ efficiencies might be gained when we more closely examine individual strategies applied in auctions, which is left for Appendix B. The analysis there shows that participants used different strategies in the auctions under study. Responding rationally to the specific rules of the all-pay auctions, they did offer higher management fees there, compared with the first-score auction. Nevertheless, unlike in the all-pay auctions, the losing participants of the first-score auction were not obliged to keep the fees on the levels bid, which resulted with the higher value of the weighted management fee in the post-auction periods.