The contextualized experimental protocol building

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The two versions of the game

Two versions of this game were tested the one after the other with the same 6 students. The difference between the two versions lies on the mechanism of water allocation followed by the dam manager and the uncertainty in the decision made by the farmer.
In the first version , the dam manager allocated the water required by the farmers with the only constraint represented by the maximum capacity of the dam. The respect of the domestic and ecological “reserve” was not compulsory. The farmers did not know how the dam manager would allocate water. The only rule the farmers knew was that there was a priority order and that they probably would not be the first to be supplied. Therefore, the farmers chose how much additional area to cultivate without knowing if they would obtain the whole amount of water needed to irrigate it. The farmers had to choose if and how much to increase the irrigated area under uncertainty.
In the second version, the dam manager’s allocation rules were transparent and known by the farmers. Unlike in the first version, the domestic and ecological reserves were respected. The allocation rule was as follows: if the total water required by the three farmers was lower than the amount of water available (i.e. 250 000 m3), each farmer would receive exactly the amount of water required. If the total water required was higher than the amount of water available, the dam manager distributed the water available among the three farmers proportionally to their original demand.
For example, if the requests for farmers 1, 2 and 3 were respectively 50 000 m3, 100 000 m3 and 150 000 m 3, the total amount of water required is equal to 300 000 m3 (which is higher than the 250 000 m3 available). Consequently, the dam manager would divide the 250 000 m3 available proportionally to their requests among the three farmers. Therefore the amounts allocated would be 41 666 m3, 83 333 m3 and 125 000 m3 for farmers 1, 2 and 3 respectively.
Unlike in the first version, the farmers required first water from the dam, and then they were informed on the amount they would receive. Only at this point the farmers decided whether or not to extend their cultivated area. Farmers’ choice regarding the extension of cultivated areas took place with no uncertainty in this version.
The following table (table 8) summarizes the common framework for the two versions of the experiment. The differences between the two versions are highlighted at the bottom of the table. A detailed description of the test phases is provided in annex B6.

General observations

The first version of the experiment lasted 81 minutes (1h21) and the second one 61 minutes (1h01) (cf. details in annex B7). The participants did not receive monetary rewards at the end of the test. Generally the participants understood quickly and completely the game. The Excel program proved to be a useful tool to help the players in their decision-making process. There were no problems with the numbers or the calculations. Only one mistake was observed. The players also allocated properly (i.e. as predicted in the model) water within coalitions, considering the different production (response) functions of each farmer.
The CGT model makes the assumption of rationality of the players, which are profit takers. By comparing the results of the test session with the « theoretical » results obtained through the model (see tables in annexes B10 and B11), it can be observed that in both versions of the experiment, unlike in the theoretical model, the result was not super-additive23, but for different reasons.

Non super-additivity analysis

In the first version, the dam manager’s criteria for the water allocation were unknown by the players, which made their decisions under uncertainty. The players asked for water even when playing as singletons. They were either optimistic on the probability of receiving water, despite the indications contained in the instructions, or, if they believed in the instructions, they did not expect the other players to ask for the full amount of water, or they went for a free riding of the reserve. Actually, only one player (farmer 3) out of three asked for the maximum amount of water. As a consequence, the domestic and ecological reserve was always preserved, except when the coalition by the farmers 1 and 2 played. In the partial coalitions phase, the farmers increased progressively their water (and surface) demand from less than expected in the theoretical model (63 700 m3) to all the water available (250 000 m 3). This result could be due to a learning effect appearing through the replication of the same situation (Loewenstein, 1999; Eber and Willinger, 2005).
Because, unlike what the theoretical model predicted (cf. annex B10), singletons and partial coalitions attempted anyway to ask for water, and the dam manager allocated what they asked for, it happened that v(1)+v(2)>v(1,2), v(1)+v(3)>v(1,3) and v(2,3) + v(1) > v(1,2,3), making the game not super-additive.
In the second version, if the total demand was higher than the 250 000 m3 available, the dam manager distributed the 250 000 m3 among the players proportionally to their request. Therefore, water allocation to each player depended also on what the other players required. The reserve was always preserved.
Unlike the assumption of rationality of the model, most players did not ask the total amount of water they could, even if in this case they did not risk anything by asking the maximum. Therefore, due to the water allocation rule, those who asked more water in a phase or round got proportionally more than those players that played conservative (e.g. farmer 3 required more than farmers 1 and 2). For the same reason the partial coalition {1,3}, asking the maximum and being faced with a low request from farmer 2, was able to have a profit v(1,3) that summed to v(2) is higher than v(1,2,3). For this reason, the second version results were not super-additive.

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Payoff sharing

During the second phase of the first version, players were asked, after they played the partial coalition round, how they would share the payoff obtained. This is not relevant information to compare with the theoretical results, but it represented complementary information. In the three cases, unlike the logical expectation of payoff sharing that follows the different productivity of the three farmers, the players chose an equal allocation between the two of them (50% each).
23 Let N be a finite set of n players in a transferable utility game. Let S and T be subsets of N (S and T are coalitions). Let v be a real-valued function defined over all the subsets of N. In the present experiment, v is the payoff obtained.
During the final debriefing after having conducted the two versions of the game, the experimenter asked why the participants chose such a sharing. Farmers 1 and 2 who had a worse response function than farmer 3 had an interest in the equal sharing. The question was to know why farmer 3 accepted such repartition. Farmer 3 did not answer. It seemed that the two players playing farmer 3 role did not remember why they chose the equal repartition. They might have considered the equal sharing as a fair choice without having assessed the different farmers’ contribution to the final profit.
The experimenter only communicated the payoffs obtained in the second version. Once the players knew all the profits obtained in the three phases they were asked whether they wanted to be in the grand coalition and, if yes, to share the grand coalition payoff. The following table (table 9) presents the results obtained in the second version.

Table of contents :

Chapter 1: Literature review
1.1. Experimental Economics and Companion Modelling
1.2. Cooperative Game Theory and Experimental Economics
1.2.1. What is Experimental Economics (EE) ?
1.2.2. Cooperative Game Theory (CGT) and EE
1.3. Methodology of EE
1.3.1. The protocol and the instructions
1.3.2. The main methodological issues of EE
Chapter 2: The CGT Model
2.1. The CGT Model* (V2)
2.2. Calibration of the model and expected results
Chapter 3: The contextualized experimental protocol building
3.1. The Experiment
3.2. First Test Session – 12th of June 2007
3.2.1. The two versions of the game
3.2.2. Results
3.2.3. Lessons
3.3. Second Test Session – 18th of July 2007
3.3.1. Differences between the two test sessions
3.3.2. Results
3.3.3. Lessons and perspectives
Chapter 4: Conclusion and research perspectives
4.1. The decontextualized experiment
4.2. The players

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