Individual and strategic behaviors in a dynamic extraction problem: results from a within-subject experiment in continuous time

ABSTRACT

We conduct a laboratory experiment to test a continuous-time model that represents a dynamic groundwater extraction problem in an infinite horizon. We compare the observations to the equilibrium path of the usual behaviours, for the case where the player is alone in extracting the resource (optimal control) and when two players extract the same resource simultaneously (differential game). We use a within-subjects design. This allows us to identify individual profiles of players playing alone and then characterize groups based on their composition with respect to these individual behaviours. We find that approximately a quarter of the players and groups succeed in playing (significantly) optimally, and none behave myopically. Moreover having an agent that behaved optimally in the control in the pair increases the likelihood that the group cooperates. We also identify other categories of players and groups that allows us to classify an additional 50% of the observations.

KEYWORDS:

• Differential games
• dynamic optimization
• experimental economics
• renewable resources
• applied econometrics

JEL CLASSIFICATION:

• C01
• C73
• C91
• C92
• Q20

Acknowledgment

We would like to thank all the persons who, at different stages of this research, helped us to improve its content through very fruitful discussions and advice. We especially thank the participants of the French Experimental Economics Association conference (2018), the International Symposium on Dynamic Games and Applications (2018), and the GREEN-Econ Spring School in Environmental Economics (2019). We also thank the French National Research Agency for its financial support (ANR GREEN-Econ, 2016-2020, grant number: ANR-16-CE03-0005).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ Simon and Stinchcombe (1989) defined, in a [0, 1] time interval, a finite set of agents and imposed some limitations on the decisions players could change. This allowed agents to play games in continuous time in the limit as the interval approaches zero. See Calford and Oprea (2017) for a laboratory implementation of the timing game developed in Simon and Stinchcombe (1989).

² Janssen et al. (2010) and Cerutti (2017) also run experiments in continuous time in the laboratory, but without an underlying theoretical model. Their objective is to implement renewable resources along both spatial and temporal dimensions. Their experiments are conducted in real time to simulate the real-life conditions of ecological systems.

³ We use a simple “bathtub” model to describe the groundwater extraction.

⁴ We omit the subindex $t$ when it is unnecessary.

⁵ The subject pool is managed by the ORSEE platform (Greiner 2015), and has about $3000$ volunteers.

⁶ A screenshot of the decision screen is provided in the Appendix.

⁷ The experimental protocol is not sufficiently detailed in the study to confirm that a procedure similar to ours was applied, in particular for computations between two instants.

⁸ Another way to implement the infinite horizon in experimental economics is to set a probability that the current period is the final one (e.g. Suter et al. 2012; Vespa 2020). With this approach, the subject does not know exactly when the repetitions will end. However, this method has two drawbacks. First, if the probability is defined individually or even per experimental session, it implies different endings to the game, which means a different number of decisions and a different history. This complicates analysis and comparison. Second, it may be interpreted by the subject as an unknown end rather than an infinite horizon.

⁹ Other rules would have been possible, such as providing the remaining available resource or, in the two-players game, dividing the remainder equally or proportionally to the quantity requested. This rule was chosen because it is easy to implement in the lab and because setting an allocation rule for the extraction in proportion to the available resource would have led to a multiplicity of equilibria, which would have greatly complicated the empirical strategy needed to compare lab results to equilibrium paths without revealing any (particularly) interesting information on the behaviour of agents..

¹⁰ Full experimental instructions are available in the online supplementary materials.

¹¹ The return flow coefficient is the quantity of water returning to the groundwater after each extraction.

¹² To take a concrete example, instead of comparing the player’s extraction $w(t)$ to the conditional constrained myopic and conditional optimal extraction, $w(t{)}_{my}^c$and $w(t{)}_{op}^c$, we could compare it to the temperature in Moscow and Istanbul, and we would find that our player’s extraction is closer to the temperature in Moscow or in Istanbul, because one MSD will always be smaller than the other, even if completely irrelevant.

¹³ An alternative is proposed by Suter et al. (2012), who run a similar regression (without the constant term) and consider that a player follows a given behaviour if the coefficient is not significantly different from $1$. A natural way to do this is to implement a Wald test with:

$\left\{\begin{array}{c}{H}_0:{\beta }_1=1\\ H_A:{\beta }_1\ne 1,\end{array}andW=\frac{{({\hat{\beta }}_1-1)}^2}{\mathrm{v}\mathrm{a}\mathrm{r}({\hat{\beta }}_1)}\to F_{{ }^{(1,300)}}\right.$

In this case, a very imprecisely estimated coefficient ${\beta }_1$ (very large $var({\hat{\beta }}_1)$) will lead us to reject $H_A$ and classify the player as myopic or optimal, while they follow neither an optimal nor myopic path. This is the reason we propose the alternative classification rule.

¹⁴ We present regression results using 5 lags. Results using 1 and 10 lags are available upon request.

¹⁵ Remember that the total payoff is the sum of the discounted payoff at each instant plus the continuation payoff.

¹⁶ The feedback representation is obtained when the solution is written according to the state variable, instead of according to time.

Additional information

Funding

The work was supported by the Agence Nationale de la Recherche [ANR-16-CE03-0005]