2.1. The price-generating mechanism

As previously mentioned, there are two assets in the market. The risk-free asset pays a fixed interest rate $r \gt 0$ per period. The infinitely lived risky asset pays a dividend $y_t$ in period $t$. The dividend is IID with an expected value $\bar{y}$. The price of the risky asset, $p_{t}$, is endogenously determined by the asset’s aggregate demand and supply. All traders are assumed to be myopic mean-variance maximizers with full information about the dividend process. The model can be solved for the market-clearing price $p_t$, see Appendix A. There we show that traders’ demand for the risky asset is an increasing linear function of the expected return $p_{i,t+1}^{e}+\bar{y}-(1+r)p_t$, where $p_{i,t+1}^{e}$ represents trader $i$’s forecast for the price in period $t+1$, made at the beginning of period $t$, prior to the realization of the market-clearing price $p_t$. In the experiment, participants’ task is to provide these price forecasts to their trader (pension fund).

In each experimental market, there are six large pension funds, each advised by a participant, and a fraction $n_t\in[0,1)$ of robot traders. These robots generate forecasts representing the fundamental price of the risky asset, defined as the discounted expected value of all future dividends, and are therefore similar to “value investors” in financial markets. For our IID dividend process, the fundamental price is

(1)\begin{equation} p^{f}=\frac{\bar{y}}{r}\,. \end{equation}

The market-clearing price in period $t$, depending on market composition, is

(2)\begin{equation} p_t = \frac{1}{1+r} \Big( (1-n_t) \bar{p}_{t+1}^{e} + n_t p^f + \bar{y} \Big)\,, \end{equation}

where $\bar{p}_{t+1}^{e}$ is the average expected price for period $t+1$ among human participants who submitted forecasts during period $t$. Substituting Eq. (1) yields

(3)\begin{equation} p_t-p^f = \frac{1-n_t}{1+r} \left( \bar{p}_{t+1}^{e}-p^f \right)\,. \end{equation}

Hence, the price deviation from the fundamental value in period $t$ depends positively on the expected deviation in period $t+1$. This creates positive expectations feedback between predicted and realized prices: if traders expect the price to rise, they buy the asset to benefit from anticipated capital gains, driving up the market-clearing price. As the right-hand side of Eq. (3) decreases with $n_t$, robots, acting as fundamentalists, mitigate this feedback. Following Hommes et al. (Reference Hommes, Sonnemans, Tuinstra and van de Velden2005), we assume these fundamentalists become more active as the price deviates further from its fundamental value, setting

(4)\begin{equation} n_t = 1-\exp \left( -\frac{1}{20} \left| \frac{p_{t-1}-p^f}{p^f} \right| \right) \,. \end{equation}

As a result, the robots’ influence grows with mispricing, acting as an endogenous mechanism to prevent bubbles from growing indefinitely. Specification (4) ensures that the mispricing must be substantial for the robots to become relevant. Even when the price is twice the fundamental value, the robots’ weight remains below 5%, which is less than one-third of the weight of each human participant’s forecast.Footnote ⁵

2.2. Experimental procedure and software

Upon arriving at the lab, participants receive both paper and on-screen instructions detailing the market environment, task, experiment structure, and payoff determination (see Online Appendix C). Following standard LtF experimental protocols (Hommes, Reference Hommes2011), participants are not explicitly provided with equations (2) and (4). Instead, they are informed that several funds allocate their wealth between a risky and a risk-free asset with the price of the risky asset determined by the equilibrium between the aggregate demand of all funds and a fixed supply. Participants act as financial advisors tasked with forecasting the future price of the risky asset as accurately as possible. They are also informed that higher price forecasts increase their fund’s investment in the risky asset, while other funds in the market are either advised by fellow participants or follow a fixed investment strategy.

The instructions explain that the experiment consists of two phases of price forecasting, followed by a brief questionnaire. In each phase, participants predict prices for a large number of consecutive periods, ranging from 120 to 180. This range is specified to discourage strategic behavior that might occur toward the end of each phase. Participants are informed of limited time to make each forecast and possible ‘waiting’ time between decisions. They are also told that waiting time, decision time, and market parameters remain constant within each phase but may vary between the two phases.

Earnings are based on participants’ forecasting accuracy and are determined as follows. At the end of the experiment, the computer program randomly selects 10 periods from each phase for payment. The total points accumulated during these 20 periods determine each participant’s payoff. For a selected period $t$, the points earned by participant $i$, $e_{i,t}$, depend on their absolute prediction error, according to

(5)\begin{equation} e_{i,t}=\frac{200}{1+ \left| p_{t}-p_{i,t}^{e} \right |}\,, \end{equation}

where $p_{i,t}^e$ is the participant’s forecast for the realized price $p_{t}$. If no forecast is made in period $t$, $e_{i,t}=0$. The hyperbolic scoring rule (5) strongly incentivizes accuracy by penalizing even small errors. A graphical representation of this scoring function, included in the instructions, can be found in Online Appendix C.

This payoff structure ensures participants have a strong incentive to be accurate in each period, regardless of their performance in other periods. Total points earned are converted at a rate of one euro per 100 points at the end of the experiment. Participants can earn between 0 and 2 euros per period, with a maximum of € $40$ for the entire experiment. In addition, each participant receives a € $10$ show-up fee.

After reading the instructions and completing a short comprehension test, participants engage in a practice round to familiarize themselves with the software. Figure 1 shows an example of the interface. Participants can submit forecasts by either typing a number into the submission box at the top of the screen or clicking directly on the main graph.Footnote ⁶ When using the graph, the selected value appears both as the forecast in the graph and in the submission box. Participants may revise their forecasts as often as they wish within the decision time. Once the decision time expires, the forecast in the submission box is automatically submitted, except in treatment $\mathbf{HLS}$ (see Section 2.3), where participants must finalize their forecast by pressing the blue ‘Submit’ button next to the submission box or the ‘Enter’ key on the keyboard.

Figure 1.

An example of the computer screen. The graph shows past forecasts (blue) and past prices (red). The table provides the same information, and also displays the “Potential earnings”, i.e., the points awarded for each period if that period is selected for payment, as computed using Eq. (5). A forecast can be entered either by typing a number into the box at the top center of the screen or by clicking on the graph in the lower part. This example is from treatment $\mathbf{HLS}$, where participants must either press the ‘Enter’ key on the keyboard or click the blue ‘Submit’ button at the top of the screen to submit their forecast

Our sessions consist of 6, 12, or 18 participants. Once all participants have completed the practice round, the computer randomly forms groups of six, and the first phase begins. The starting screen announces the interest rate $r$, the mean dividend $\overline{y}$, and the waiting and decision times. This information remains visible at the top of the screen throughout the phase, along with a countdown timer. The realized price for period $t=1$ is determined by participants’ forecasts for period $t=2$. Thus, participants begin by submitting forecasts for periods 1 and 2. No robots are active in period 1. To limit initial forecast dispersion, the instructions state that the first two prices in the first phase are “likely to lie between 0 and 200,” and the first two prices in the second phase are “likely to lie between 0 and 100.” These ranges include the fundamental price. Subsequently, in each period $t \gt 1$, participants submit a price forecast for period $t+1$.

The computer interface includes a graph displaying the participant’s forecasts (blue) and realized prices (red) for the most recent $20$ periods (Figure 1). A smaller graph in the upper-left corner shows the entire time series from the start of the phase, with both graphs automatically adjusting their vertical axes to accommodate higher prices or forecasts.Footnote ⁷ A table on the right-hand side shows the participant’s forecasts, realized prices, and the number of points per period, computed using Eq. (5). These points are labeled “Potential Earnings” because they are only earned if the corresponding period is selected for final payment.

At the end of the phase, participants are notified of their transition to a new market. They are randomly re-matched into groups of six, and the second phase begins with new market parameters, including new waiting and decision times.

After finishing the two phases, participants complete the standard three-question Cognitive Reflection Test (Frederick, Reference Frederick2005). They then fill out a questionnaire on demographic information (e.g., gender, age, study program), their previous experience with lab experiments and the forecasting strategies they used in each of the phases in the experiment (see Online Appendix D). Finally, the computer informs each participant which 20 rounds were chosen for payment and displays their final payoff. All payments are made privately.

2.3. Treatments

We solicit forecasts from participants under two distinct conditions: the low time pressure (LTP) condition and the high time pressure (HTP) condition. These conditions differ in the amount of time participants are given to submit their forecasts.

Each decision period consists of two stages: a waiting time and a decision time. During the waiting time, participants can view the computer interface, which includes a graph and table displaying past prices and forecasts. They can also navigate the screen with their mouse and select potential forecasts. However, the submission box does not appear until the waiting time ends and the decision time begins, at which point it displays the last value clicked and allows participants to submit their forecasts.

In the LTP condition, the waiting time is set to 10 seconds, followed by a decision time of 15 seconds, totaling 25 seconds per period. This duration aligns with the average time taken for forecasts in previous LtF experiments.Footnote ⁸ In contrast, the HTP condition features no waiting time and a decision time of only $6$ seconds, placing participants under substantial time pressure. This impact was confirmed by responses to the post-experiment questionnaire.

Each participant experienced both conditions, implemented in two separate phases. However, this design was primarily intended to ensure comparable session lengths and participant payoffs, as encouraged by the CREED-lab guidelines, rather than to facilitate within-subject comparisons. We do not primarily focus on within-subject comparisons due to the potential for strong order effects, which are beyond our control. Individual experiences in LtF experiments heavily depend on group composition, which, as demonstrated by Hennequin (Reference Hennequin2018), can significantly influence subsequent behavior. Consequently, we anticipated that the second-phase data would be of lower quality, as heterogeneous prior experiences would likely have a pronounced impact. Additionally, variations in the show-up rates made it impossible to guarantee comparable rematching for sessions conducted on different days.

For these reasons, our design incorporates between-subjects treatments. The treatments differ in the order in which participants experience the two conditions. To isolate the time pressure effect as sole source of any between-treatment differences, we used identical market parameters across treatments in the first phase. To mitigate order effects in the second phase, we modified the parameters to generate a substantially different fundamental price ( $p^{f}=126.4$ in the first phase and $p^{f}=71.2$ in the second phase), which remained constant across treatments. To mitigate potential end-of-phase effects caused by round numbers of periods or reusing the same number of periods, we predetermined the total number of periods for each condition: 146 for the LTP condition and 159 for the HTP condition. However, participants were only informed about the range of possible periods. The selected number of periods was also guided by the need to ensure that sessions could be completed within approximately two hours.

We refer to the treatment that begins with the LTP condition as treatment $\mathbf{LH}$ and to the treatment that begins with the HTP condition as treatment $\mathbf{HL}$. Table 1 summarizes the treatments, specifying their parameters, the number of markets, and their notation. The letter indicates the corresponding time pressure condition. Markets in the second phase are denoted with a superscript 2. While the number of markets for each treatment is the same in the first and second phases, their composition differs due to rematching. We conducted 12 markets in each phase of treatment $\mathbf{LH}$ and 10 markets in each phase of treatment $\mathbf{HL}$.

Table 1.

Overview of the treatments. The last two columns display the experimental market notations and the parameter values for each of the experiment’s two phases. In both phases, the interest rate is $r\!=\!0.05$. Each market consists of six human participants

During the initial sessions of our experiment, we observed a series of downward spikes in individual forecasts under the HTP condition (e.g., panels 4, 8, and 10 in Figures F4 and F5, Online Appendix F). These anomalies appear to stem from participants being unable to complete their forecasts in the submission box within the allotted time.Footnote ⁹ Such outliers, absent under the LTP condition, may have lasting effects on price dynamics. To rule out these outliers as the primary driver of differences between time pressure conditions, we introduced an additional treatment, $\mathbf{HLS}$, as a robustness check. This treatment was identical to $\mathbf{HL}$ in all respects, except that participants were required to explicitly confirm their forecasts by either pressing the “Enter” key or clicking the “Submit” button added next to the submission box. We conducted nine markets in each phase of treatment $\mathbf{HLS}$.

Finally, it is important to note that in all treatments, the average forecast used in Eq. (2) was calculated based only on submitted forecasts. Consequently, for some periods the average forecast could be based on fewer than six forecasts if not all participants managed to submit their forecasts for those periods within the time limit.

2.4. Hypotheses

The first research question examines whether the emergence of large bubbles and crashes in asset prices, frequently observed in LtF experiments, is a transient phenomenon. That is, will market prices eventually stabilize and converge to their fundamental value? Existing LtF experiments provide limited evidence on this issue, as they typically span only about 50 periods, leaving it unclear whether price volatility decreases or increases over time.

It is plausible to hypothesize that, given sufficient opportunities to learn within the stationary environment of the experiment, participants may improve their predictions, ultimately driving market prices toward their fundamental value. Moreover, participants’ earnings, which depend on absolute forecast errors, tend to be substantially lower in volatile markets than in those where prices closely track the fundamental value. This creates a strong incentive for participants to adapt their prediction strategies when such strategies contribute to pronounced volatility.

This leads to the following hypothesis:

The second research question focuses on the effect of time pressure on price volatility and mispricing. The direction of this effect is not evident a priori. On the one hand, time pressure has been shown to reduce decision-making quality (Kocher and Sutter, Reference Kocher and Sutter2006) and increase market volatility in experimental asset markets (Ferri et al., Reference Ferri, Ploner and Rizzolli2021). Under increased time pressure, participants have less opportunity for thorough deliberation, which may lead to forecasting errors and, consequently, heightened price volatility and mispricing.

This consideration leads to the following hypothesis:

On the other hand, bubbles in earlier LtF experiments appear to be driven by participants coordinating on a trend-extrapolating prediction strategy, where forecasts depend on the last two observed prices. When all participants in a market extrapolate trends from past prices, the positive feedback inherent in the underlying price-generating mechanism amplifies the trend, giving rise to large bubbles in asset prices. Under increased time pressure, however, coordination on a common prediction strategy may become more difficult, potentially inhibiting the emergence of large bubbles and crashes. For example, Anufriev et al. (Reference Anufriev, Chernulich and Tuinstra2022) found that increasing task complexity—such as by extending the forecasting horizon in LtF experiments—hinders the coordination of expectations and results in greater price stability. Thus, rather than increasing volatility, higher time pressure might actually promote more stable price dynamics.

This consideration leads to an alternative version of the second hypothesis:

We will now discuss the experimental results in view of Hypotheses 1, 2, and 3.

3.1. Market prices

In this section, we analyze the market prices using the first-phase data. The top panel of Figure 2 corresponds to the LTP condition and shows the prices (thin gray lines) for each $\mathbf{L}$ market (i.e., from the first phase of treatment $\mathbf{LH}$). The remaining two panels correspond to the HTP condition, illustrating the prices for $\mathbf{H}$ and $\mathbf{HS}$ markets (i.e., from the first phase of treatments $\mathbf{HL}$ and $\mathbf{HLS}$). In each panel, the black line represents the median price across all markets in each period.Footnote ¹⁰ To facilitate comparisons, the vertical axis in all panels ranges from 0 to 500, which is approximately four times the fundamental price. Note that in some periods, particularly in treatment $\mathbf{LH}$, market prices exceed 500.Footnote ¹¹

Figure 2.

Median prices (thick black line) and prices in individual markets (gray lines) during the first phase of the three experimental treatments. The fundamental price, $p^f=126.4$, is indicated by the dashed horizontal line

We can make several preliminary observations by comparing the median prices in Figure 2, further supported by the prices in each market. First, median prices under the LTP condition show significantly higher overvaluation compared to those under the HTP condition. In contrast, there appears to be no clear difference between the median dynamics of $\mathbf{H}$ and $\mathbf{HS}$ markets, both conducted under the HTP condition. This suggests that differences between time pressure conditions cannot be attributed to participants submitting incomplete predictions in treatment $\mathbf{HL}$.

Second, price volatility is notably high under the LTP condition, with prices oscillating wildly in all markets (see also footnote 11). Conversely, price volatility under the HTP condition is considerably lower. Most markets in the lower two panels exhibit relatively minor oscillations and tend to converge quickly to the fundamental value (e.g., markets $\mathbf{H}$3, $\mathbf{H}$4, $\mathbf{HS}$1, $\mathbf{HS}$3 and $\mathbf{HS}$4).

Third, under the LTP condition, oscillations begin almost immediately, while under the HTP condition, they seem to be more prevalent in the second half of the experiment, following an initial phase of relatively minor volatility (e.g., markets $\mathbf{H}$5, $\mathbf{H}$8, $\mathbf{HS}$8 and $\mathbf{HS}$9). Moreover, fluctuations under the LTP condition seem to diminish over time, though this tendency is fragile; in some markets, oscillations re-emerge after prices seem to have converged (e.g., markets $\mathbf{L}$1, $\mathbf{L}$2, $\mathbf{L}$8 and $\mathbf{L}$9). By contrast, under the HTP condition, there is no evident structural decline in price volatility over time.

To corroborate these observations, we consider two quantitative measures: the interquartile range (IQR) to evaluate price volatility and the median of the relative absolute deviations (RAD) from the fundamental value to assess mispricing.Footnote ¹² Figure 3 shows the IQR (left panel, logarithmic scale) and the median RAD (right panel) for all 31 experimental markets of the first phase. Each panel is divided into three sections corresponding to the $\mathbf{L}$, $\mathbf{H}$, and $\mathbf{HS}$ markets. Within each section, the statistics for each market are computed over three different time periods: 11-50 (blue dots on the left), 1-145 (black dots in the middle), and 106-145 (red dots on the right), with disks indicating the median value. Numerical values for both measures for all markets are provided in Table B1 in Appendix B.

Figure 3.

Measures of price volatility (IQR, left panel, logarithmic scale) and mispricing (Median of RAD, right panel) by treatment for each market of the first phase, computed over three different time periods: $11$– $50$ (blue dots), $1$– $145$ (black dots), and $106$– $145$ (red dots). The disks show the median over the markets

Figure 3 illustrates that, under the LTP condition, both measures tend to decrease over time. Notably, as shown in Table B1, the IQR decreases in all $\mathbf{L}$ markets except one ( $\mathbf{L}$12) and the median RAD decreases in all $\mathbf{L}$ markets except two ( $\mathbf{L}$8 and $\mathbf{L}$12).Footnote ¹³

This tendency is statistically confirmed by a Wilcoxon signed-rank test; see Table 2 for the $p$-values of tests presented in this section. Comparing $\mathbf{L}$ markets between periods 11-50 and 106-145, the $p$-values for the two-sided tests are 0.0010 for the IQR and 0.0210 for the median RAD. These results indicate that, under the LTP condition, both price volatility and mispricing are significantly lower in later periods compared to initial periods, at the 1% and 5% significance levels, respectively. In contrast, in markets conducted under the HTP condition ( $\mathbf{H}$ or $\mathbf{HS}$), neither price volatility nor mispricing shows a statistically significant decrease between periods 11-50 and 106-145.

Table 2.

$p$-values of the corresponding tests (see the last column) for various comparisons on the first-phase data

Note: The first two columns specify the markets and time periods for the data. Asterisks

* , ** and *** indicate $p$-values that are below $10\%$, $5\%$ and $1\%$, respectively.

Consequently, we reject Hypothesis 1 for the first-phase data from treatments $\mathbf{HL}$ and $\mathbf{HLS}$, but not for treatment $\mathbf{LH}$.

Recall that in earlier LtF experiments, prices do not converge to their fundamental value within the standard duration of 50 periods. This aligns with the dynamics observed in our $\mathbf{L}$ markets: until period 50, all twelve markets exhibit large price fluctuations. However, as Result 1 indicates, price fluctuations tend to decrease after these initial 50 periods. Long-run dynamics differ significantly from those at the outset, though boom-and-bust cycles can occasionally re-emerge. We conclude that while bubbles and crashes initially seem persistent, they are ultimately transient in this stationary environment and diminish over time.

Turning to the comparison of time pressure conditions reveals striking differences in price patterns during the first 50 periods. In particular, the HTP condition (covering $\mathbf{H}$ and $\mathbf{HS}$ markets) exhibits smaller price fluctuations than the LTP condition and earlier LtF experiments. For example, among the 19 HTP markets, only two ( $\mathbf{H}$6 and $\mathbf{HS}$5) show larger fluctuations (measured by the IQR) in periods 11–50 than the most stable LTP market ( $\mathbf{L}$8); see Table B1. Moreover, when price fluctuations arise under the HTP condition, they are often triggered by extreme predictions from individual participants.Footnote ¹⁴ In contrast, such outliers are rare during the first 50 periods of the LTP condition, indicating that price fluctuations there cannot be attributed solely to idiosyncratic behavior by individual participants.

Hence, increased time pressure seems to inhibit price volatility and the emergence of bubbles and crashes during the initial 50 periods. This is confirmed by comparing the IQR and the median RAD for periods 11-50 using a two-sided Mann-Whitney-Wilcoxon (MWW) test. Differences between the $\mathbf{L}$ and $\mathbf{H}$ markets, and between the $\mathbf{L}$ and $\mathbf{HS}$ markets, are statistically significant at the 1% level (see Table 2).

Figures 3 and B1 to B3 suggest that the effect of time pressure on price behavior diminishes toward the end of the phase. For instance, although the median RAD is on average still higher under the LTP condition than under the HTP condition, the mean IQR for periods 106-145 is lower under the LTP condition, with six of the seven markets exhibiting the highest IQR belonging to the HTP condition (Table B1). While the differences in the IQR between the LTP and HTP conditions are not statistically significant for periods 106-145, the differences in the median RAD are significant at the 10% level.

This leads to our second result, confirming Hypothesis 3.

Finally, test results reported in the lower part of Table 2 indicate that, regardless of the periods examined, there is no significant difference in either IQR or median RAD between the first-phase markets conducted under the two HTP treatments ( $\mathbf{HL}$ and $\mathbf{HLS}$). Thus, incomplete predictions have a minimal impact on price dynamics.

3.2. Robustness of results 1 and 2

In this section, we first assess the robustness of Results 1 and 2 by using the entire dataset, including second-phase data. We subsequently discuss whether non-responses by participants in the HTP condition could have influenced our findings.

3.2.1. Tests using pooled data

Table 3 presents the $p$-values for tests applied to the entire experimental dataset. Specifically, we pool all 31 markets under the LTP condition (comprising 12 $\mathbf{L}$ markets from the first phase, and 10 $\mathbf{L}^2$ and 9 $\mathbf{LS}^2$ markets from the second phase) and all 31 markets under the HTP condition (comprising 10 $\mathbf{H}$ and 9 $\mathbf{HS}$ markets from the first phase, and 12 $\mathbf{H}^2$ markets from the second phase).

Table 3.

$p$-values of the corresponding tests (see the last column) for comparisons based on all data

Note: The first two columns specify the markets and time periods for the data.

* , ** and ***indicate $p$-values that are below $10\%$, $5\%$ and $1\%$, respectively

The upper part of the table supports Result 1: the differences in both volatility (IQR) and mispricing (median RAD) across time are highly statistically significant for the LTP condition, reflecting the long-run stabilization observed in these markets. For HTP markets, as in the first-phase data, there is no significant difference in the IQR, though the median RAD now differs at the 5% significance level.Footnote ¹⁵

The lower part of Table 3 supports Result 2: both volatility and mispricing differ significantly across time pressure conditions for periods 11-50 at the $1\%$ level. Such strong significant differences are not observed for the ending interval or for all periods.

3.2.2. Effect of non-submissions

In all three treatments, participants occasionally fail to submit forecasts, resulting in prices being based on fewer than six human forecasts. Could these “non-submissions” explain the observed results?

Figure 4 illustrates the time evolution of the fraction of non-submitted forecasts during both phases, calculated across all markets within each treatment and smoothed using a 5-period moving average. Thin solid lines represent non-submissions in the $\mathbf{LH}$ and $\mathbf{HL}$ treatments, and thick solid lines represent those in the $\mathbf{HLS}$ treatment. Blue lines correspond to the HTP condition, while red lines represent these fractions under the LTP condition.

Figure 4.

The fraction of participants who did not submit a forecast in a given time period, shown as a 5-period moving average across all markets over two phases. Red lines represent the LTP condition, and blue lines represent the HTP condition. The phase change, occurring after period 146 in the $\mathbf{LH}$ treatment and after period 159 in the $\mathbf{HL}$ and $\mathbf{HLS}$ treatments, is indicated by the vertical dashed lines

Overall, the fraction of non-submissions is higher in HTP markets, likely due to elevated time pressure, as reflected by between-phase jumps.Footnote ¹⁶ Non-submissions may affect price dynamics through a selection effect if participants with more non-submissions differ in forecasting accuracy from others (see Kocher et al., Reference Kocher, Schindler, Trautmann and Xu2019 for a discussion of selection effects induced by time pressure in decision-making).

To rule out this possibility, we conduct two tests. First, despite the higher number of non-submissions in $\mathbf{HS}$ markets, there are no significant differences in IQR or median RAD compared to $\mathbf{H}$ markets (see Table 2). Second, we assess correlations across all HTP markets between the number of non-submissions in periods 1–10 and the IQR and median RAD in periods 11–50. None of these correlations is statistically significant.Footnote ¹⁷ These findings indicate that the impact of higher time pressure on price dynamics cannot be attributed to differences in non-submission rates.

Figure 4 also suggests that, despite the many decision periods, fatigue was not a substantial factor in our experiment. If fatigue had been significant and the incentives insufficient to counteract it, we would expect non-submissions to increase over time, particularly toward the end of the experiment. However, this pattern is not observed.

3.3. Market expectations

In this section, we provide structural behavioral explanations for Results 1and 2. Bubbles and crashes in LtF experiments are often linked to participants’ tendency to coordinate on trend-extrapolating forecasting heuristics (Anufriev & Hommes, Reference Anufriev and Hommes2012; Hommes et al., Reference Hommes, Sonnemans, Tuinstra and van de Velden2005). We considered the possibility that the greater stability under HTP might be explained by a failure to coordinate expectations, but find no evidence to support this hypothesis (see Appendix B). An alternative explanation is that the long-run stability under LTP and the lower incidence of bubbles under HTP result from the evolution of forecasting heuristics, and how this is influenced by time pressure.

We analyze market expectations, $\bar{p}_{t+1}^e$, defined as the average forecast submitted by participants in a given market and period. Forecasting heuristics, modeled as:

(6)\begin{equation} \bar{p}_{t+1}^e = a + b_1 p_{t-1} + b_2 p_{t-2} + \nu_t = a + \theta_1 p_{t-1} + \theta_2 (p_{t-1}-p_{t-2}) + \nu_t\,, \end{equation}

where $a$, $b_1$, $b_2$ are fixed coefficients, $\nu_t$ is an error term, and $\theta_1 = b_1 + b_2$, $\theta_2 = - b_2$, have been shown to describe expectations in previous LtF experiments well (Hommes et al., Reference Hommes, Sonnemans, Tuinstra and van de Velden2005). Rule (6) extrapolates price trends when $b_2 \lt 0$, or equivalently, when $\theta_2 \gt 0$.

For each of the 62 experimental markets, we estimate the forecasting heuristic in Eq. (6) separately for periods 11–50 and 106–145. Table 4 reports the average and median estimated coefficients for each treatment and phase.Footnote ¹⁸

Table 4.

The average and median (calculated over all markets within a treatment and phase) of the estimated coefficients for the prediction rule in Eq. (6)

Table 4 indicates that forecasting rules differ structurally between the beginning and end of the phase, as well as between the low and high time pressure conditions. In $\mathbf{L}$ markets, the median estimate of $b_2$ for periods 11-50 is $-0.95$, reflecting a strong tendency to extrapolate trends: if the price rises by one unit between the two most recent periods, participants’ average forecast increases by nearly one additional unit. This trend extrapolation weakens toward the end of the phase, with the median $b_2$ estimate increasing to $-0.63$ for periods 106-145. By contrast, there is virtually no trend extrapolation in the first phase in the HTP markets ( $\mathbf{H}$ and $\mathbf{HS}$) during the initial periods, with median $b_2$ estimates of $-0.05$ for periods 11-50. The same tendencies are observed in the second phase, though trend extrapolation is initially stronger in $\mathbf{H}^2$ markets.

To visualize these tendencies, Figure 5 displays scatter plots of the estimated coefficient pairs $(b_1, b_2)$ from Eq. (6) for the 31 first-phase markets for periods 11-50 (left) and 106-145 (right). The triangle and parabola divide the $(b_1,b_2)$ space into regions with qualitatively different market dynamics generated by the prediction rule in Eq. (6).Footnote ¹⁹ Prices converge if the pair $(b_1,b_2)$ lies inside the triangle and diverge if it lies outside. Dynamics are monotone if $(b_1,b_2)$ lies above the parabola and are oscillatory if it lies below. The closer $b_2$ is to the lower horizontal edge of the triangle, the more persistent the oscillations.

Figure 5.

Scatter plots of estimated $(b_1,b_2)$ coefficients from market expectations, Eq. (6) for first-phase markets during periods 11–50 (left panel) and 106–145 (right panel). Prices converge for points inside the triangle. They oscillate below the parabola

The estimates for $\mathbf{L}$, $\mathbf{H}$, and $\mathbf{HS}$ markets are represented by dots, filled squares, and non-filled squares, respectively. For periods 11–50 (left), most coefficient pairs for $\mathbf{L}$ markets are located in the lower right corner of the triangle, often near the boundary. This pattern indicates persistent oscillations. In contrast, the estimated coefficients for HTP markets cluster above the parabola, often with an estimated value of $b_2$ close to 0, suggesting that prices converge monotonically and rapidly.

When comparing this with periods 106–145 (right), we observe how forecasting rules evolve, revealing distinct patterns between conditions. In LTP markets, heuristics shift away from the corner inside the triangle, in a stabilizing direction. Instead, in most HTP markets, estimated market expectations drift in the opposite direction, toward the lower right corner of the triangle, though a few exceptions remain where the estimated value of $b_2$ equals zero. Thus, while trend extrapolation is more pronounced under LTP compared to HTP during the initial periods, this pattern has largely disappeared by the phase’s end.

This discussion is summarized as

Three additional observations supporting this result can be inferred from Table B2 (Appendix B). First, in all 12 $\mathbf{L}$ markets, the estimated coefficient $b_2$ for periods 11-50 is significantly different from zero, and all these coefficients are negative. Second, in 11 of the 12 $\mathbf{L}$ markets, the estimate of $b_2$ is lower (in absolute value) for periods 106-145 than for periods 11-50. Third, among the 19 HTP markets of the first phase, the estimate of $b_2$ is significantly different from zero in only seven markets for periods 11-50, increasing to 12 markets for periods 106-145. In fact, in several HTP markets in the first phase ( $\mathbf{H}$2, $\mathbf{HS}$3, and $\mathbf{HS}$7), market expectations are close to naïve expectations (i.e., $\bar{p}_{t+1}^e = p_{t-1}$), for periods 11-50.

3.4. Individual expectations and adaptation

In Section 3.3, we analyzed the evolution of market expectations. In this section, we investigate individual forecasting heuristics. For each participant, and for periods 11-50 and 106-145 in both the first and second phases, we estimate the model

(7)\begin{equation} p^{e}_{i,t+1} = \alpha + \beta_1 p_{t-1} + \beta_2 p_{t-2} + \beta_3 p^e_{i,t}+ \epsilon_t\,, \end{equation}

where $p^e_{i,t}$ is participant $i$’s prediction for price $p_t$, $\alpha$, $\beta_1$, $\beta_2$, and $\beta_3$ are fixed coefficients, and $\epsilon_t$ is an error term. Eq. (7) nests several commonly used expectations models, allowing us to classify participants’ forecasting behavior.Footnote ²⁰

The results of this classification are presented in Table 5. They confirm the tendencies described in Result 3. Under the LTP condition, 92% of participants are classified as trend-extrapolative at the beginning of the experiment. However, this fraction decreases to 57% by the end of the first phase, accompanied by an increase in the fraction of (stabilizing) adaptive participants. In contrast, under the HTP condition, a substantial fraction of adaptive forecasters is observed both early on and later in the first phase.

Table 5.

Classification of participants based on their forecasting behavior. Heuristic (7) is estimated for periods 11-50 and 106-145. The table reports the fractions of participant types within each market

Because our design includes within-subject variation in time pressure, we apply the classification to verify whether the forecasting behavior of subjects changed substantially between the two phases. Table 6 presents the ‘transition matrix’, pooled across all treatments, which shows how participants transitioned between trend-extrapolating, adaptive, and unclassified behavior (rows = LTP, columns = HTP), or retained the same behavior, based on periods 11–50 in each phase. Note that some participants experienced the LTP condition before the HTP condition, and for other participants it was the other way around—all of these participants are pooled in the entries in Table 6. Table F6 in Online Appendix F provides transition matrices for each of the three treatments separately.

Table 6.

Participants’ transition matrix based on the classification of individual forecasting heuristics, derived from Eq. (7), estimated for periods 11–50 in each experimental phase. The data are pooled across all treatments, with LTP behavior in rows and HTP behavior in columns

The data show that, out of 155 participants who extrapolate trends under LTP conditions, half (77) also do so under HTP, while the other half use other behaviors, including 49 (32%) adopting adaptive behavior. In contrast, of the 85 participants using trend-extrapolation in HTP, a large majority (91%) also does so in LTP. To visually illustrate the within-subject findings, Figure F8 in Online Appendix F presents a scatter plot comparing the $-\beta_2$ estimates (the trend-extrapolation coefficient) for each participant under LTP and HTP. This figure highlights a tendency for lower extrapolation under HTP. This supports Result 3 at the individual level.

A natural interpretation of our findings regarding changes in forecasting behavior under different time pressure conditions is that participants in the LTP condition, having more time to form predictions, tend to adopt more sophisticated approaches. For example, they analyze the time series, detect patterns, and extrapolate trends when making forecasts. In contrast, under the HTP condition, predictions rely more heavily on the most recent information—the last available price—due to time constraints. Incidentally, this interpretation is supported by a textual analysis of participants’ responses to the post-experimental questionnaire, where they describe their forecasting strategies under each time pressure condition (see Online Appendix E). For example, a word frequency analysis of the responses reveals that terms like ‘trend’ were more frequently mentioned under the LTP condition. An illustrative response is: “I had enough time to look back and make forecasting decisions based on past trends.” Conversely, terms like ‘time’ were more frequently mentioned under the HTP condition. Thus, a typical response is: “I had no time to think, so I almost always filled in the previous price.” Footnote ²¹

Differences in individual forecasting behavior translate into differences in price dynamics. Under the LTP condition, trend-extrapolating heuristics are at least directionally confirmed due to positive feedback in the underlying price dynamics, Eq. (2), leading to significant fluctuations in market-clearing prices. High price volatility results in forecast errors for participants, which consequently reduces their earnings. This may prompt participants to adapt their prediction rules and avoid strong extrapolation, eventually leading to more stable prices. However, we observe that such adaptation requires a substantial number of periods.

The situation is different under the HTP condition. Decision time constraints immediately encourage reliance on the most recently observed price. When many participants in a market adopt this behavior, it leads to a relatively stable time series of market-clearing prices. Over time, as participants gain experience with the decision environment, they may adapt to the time pressure and learn to identify and extrapolate price trends. However, given the fragility of the system— specifically, the feedback coefficient being close to 1—instability may arise. Still, it might be easier for participants in the HTP condition to revert to stable rules as they observe a decline in their forecasting errors.